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8.1: Dynamics of two Interacting Species - Biology

8.1: Dynamics of two Interacting Species - Biology



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In the first part of this book you’ve seen the two main categories of single-species dynamics—logistic and orthologistic, with exponential growth being an infinitely fine dividing line between the two. And you’ve seen how population dynamics can be simple or chaotically complex.

Moving forward you will see three kinds of two-species dynamics—mutualism, competition, and predation—and exactly forty kinds of three-species dynamics, deriving from the parameters of the population equations and their various combinations.

To review, the population dynamics of a single species are summarized in the following equation.

[frac{1}{N}frac{dN}{dt},=,r,+,sN]

Here parameter r is the “intrinsic growth rate” of the species— the net rate at which new individuals are introduced to the population when the population is vanishingly sparse, and s is a “density dependence” parameter that reflects how the size of the population affects the overall rate. Parameter s is key. If s is negative, the population grows “logistically,” increasing to a “carrying capacity” of −r /s, or decreasing to that carrying capacity if the population starts above it. If s is positive, then the population grows “orthologistically,” increasing ever faster until it encounters some unspecified limit not addressed in the equation. Exponential growth is the dividing line between these two outcomes, but this would only occur if s remained precisely equal to zero.

How should this single-species equation be extended to two species? First, instead of a number N for the population size of one species, we need an N for each species. Call these N1 for species 1 and N2 for species 2. Then, if the two species do not interact at all, the equations could be

[frac{1}{N_1}frac{dN_1}{dt},=,r_1,+,s_{1,1},N_1]

[frac{1}{N_2}frac{dN_2}{dt},=,r_2,+,s_{2,2},N_2]

Here r1and r2 are the intrinsic growth rates for N1 and N2, respectively, and s1,1 and s2,2 are the density dependence parameters for the two species. (The paired subscripts in the two-species equations help us address all interactions.)

There are thus four possible si,j parameters here:

  • s1,1 : How density of species 1 affects its own growth.
  • s1,2 : How density of species 2 affects the growth of species 1.
  • s2,1 : How density of species 1 affects the growth of species 2.
  • s2,2 : How density of species 2 affects its own growth.

With these parameters in mind, here are the two-species equations. The new interaction terms are in blue on the right.

[frac{1}{N_1},frac{dN_1}{dt},=,r_1,+,s_{1,1}N_1,+,color{blue}{s_{1,2}N_2}]

[frac{1}{N_2},frac{dN_2}{dt},=,r_2,+,s_{2,2}N_2,+,color{blue}{s_{2,1}N_1}]

In the single-species equations, the sign of the s term separates the two main kinds of population dynamics—positive for orthologistic, negative for logistic. Similarly, in the two-species equations, the signs of the interaction parameters s1,2 and s2,1 determine the population dynamics.

Two parameters allow three main possibilities—(1) both parameters can be negative, (2) both can be positive, or (3) one can be positive and the other negative. These are the main possibilities that natural selection has to work with.

Figure (PageIndex{1}). Both interaction parameters negative, competition.

Competition. First consider the case where s1,2 and s2,1 are both negative, as in Figure (PageIndex{1}).

For a single species, parameter s being negative causes the population to approach a carrying capacity. The same could be expected when parameters s1,2 and s2,1 are both negative—one or both species approach a carrying capacity at which the population remains constant, or as constant as external environmental conditions allow.

One example is shown in Figure (PageIndex{2}), where the population of each species is plotted on the vertical axis and time on the horizontal axis. Here Species 2, in red, grows faster, gains the advantage early, and rises to a high level. Species 1, in blue, grows more slowly but eventually rises and, because of the mutual inhibition between species in competition, drives back the population of Species 2. The two species eventually approach a joint carrying capacity.

In other cases of competition, a “superior competitor” can drive the other competitor to extinction—an outcome called “competitive exclusion.” Or, either species can drive the other to extinction, depending on which gains the advantage first. These and other cases are covered in later chapters.

In any case, when both interaction terms s1,2 and s2,1 are negative, in minus–minus interaction, each species inhibits the other’s growth, which ecologists call the “interaction competition”.

Mutualism. The opposite of competition is mutualism, where each species enhances rather than inhibits the growth of the other. Both s1,2 and s2,1 are positive.

Depicted in Figure (PageIndex{3}) is a form of “obligate mutualism,” where both species decline to extinction if either is not present. This is analogous to a joint Allee point, where the growth curves cross the horizontal axis and become negative below certain critical population levels. If this is not the case and the growth curves cross the vertical axis, each species can survive alone; this is called “facultative mutualism,” and we’ll learn more about it in later chapters.

For now, the important point is how mutualistic populations grow or decline over time. A single species whose density somehow enhances its own rate of growth becomes orthologistic, increasing ever more rapidly toward a singularity, before which it will grow so numerous that it will be checked by some other inevitable limit, such as space, predation, or disease.

It turns out that the dynamics of two species enhancing each other’s growth are similar to those of a single species enhancing its own growth. Both move to a singularity at ever increasing rates, as illustrated earlier in Figure 4.2.1 and below in Figure (PageIndex{4}). Of course, such growth cannot continue forever. It will eventually be checked by some force beyond the scope of the equations, just as human population growth was abruptly checked in the mid-twentieth century— so clearly visible earlier in Figure 6.3.1.

Predation. The remaining possibility for these two-species equations is when one interaction parameter si,j is positive and the other is negative. In other words, when the first species enhances the growth of the second while the second species inhibits the growth of the first. Or vice versa. This is “predation,” also manifested as parasitism, disease, and other forms.

Think about a predator and its prey. The more prey, the easier it is for predators to catch them, hence the easier it is for predators to feed their young and the greater the predator’s population growth. This is the right part of Figure (PageIndex{5}). The more predators there are, however, the more prey are captured; hence the lower the growth rate of the prey, as shown on the left of the figure. N1 here, then, represents the prey, and N2 represents the predator.

Prey can survive on their own, without predators, as reflected on the left in positive growth for N1 when N2 is 0. Predators, however, cannot survive without prey, as reflected on the right in the negative growth for N2 when N1 is 0. This is like an Allee point for predators, which will start to die out if the prey population falls below this point.

The question here is this: what will be the population dynamics of predator and prey through time? Will the populations grow logistically and level off at a steady state, as suggested by the negative parameter s1,2, or increase orthologistically, as suggested by the positive parameter s2,1?

Actually, they do both. Sometimes they increase faster than exponentially, when predator populations are low and growing prey populations provide ever increasing per capita growth rates for the predator, according to the right part of Figure (PageIndex{5}). In due time, however, predators become abundant and depress prey populations, in turn reducing growth of the predator populations. As shown in Figure (PageIndex{6}), the populations oscillate in ongoing tensions between predator (red line) and prey (blue line).

Examine this figure in detail. At the start, labeled A, the prey population is low and predators are declining for lack of food. A steady decline in the number of predators creates better and better conditions for prey, whose populations then increase orthologistically at ever accelerating per capita rates as predators die out and conditions for prey improve accordingly.

But then the situation turns. Prey grow abundant, with the population rising above the Allee point of the predator, at B. The number of predators thus start to increase. While predator populations are low and the number of prey is increasing, conditions continually improve for predators, and their populations grows approximately orthologistically for a time.

Then predators become abundant and drive the growth rate of the prey negative. The situation turns again, at C. Prey start to decline and predator growth becomes approximately logistic, leveling off and starting to decline at D. By E it has come full circle and the process repeats, ad infinitum.

While Figure (PageIndex{6}) illustrates the classical form for predator-prey interactions, other forms are possible. When conditions are right, the oscillations can dampen out and both predator and prey populations can reach steady states. Or the oscillations can become so wild that predators kill all the prey and then vanish themselves. This assumes some effectively-zero value for N1 and N2, below which they “snap” to zero. Or prey populations can become so low that predators all die out, leaving the prey in peace. Or both can go extinct. Or, in the case of human predators, the prey can be domesticated and transformed into mutualists. More on all such dynamics in later chapters.


Spatial Dynamics and Ecosystem Functioning

Copyright: © 2010 Oswald J. Schmitz. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: No specific funding was received for this article.

Competing interests: The author has declared that no competing interests exist.

Classical theory of species dynamics in ecosystems is built on the concept of homogeneous, reciprocal interaction. The concept is borrowed from that branch of physics and chemistry dealing with reaction kinetics of molecules in well-mixed gases and liquids. It idealizes individual entities—no longer molecules but now individuals of a species—as interacting with each other or with their predators or competitors in such a way that each individual has an equal likelihood of interacting with every other individual in the system. There is no spatial structure in the system in fact, space is assumed to be immaterial to system dynamics.

But, any keen observer of nature may cry foul. Unlike the simplified theoretical conception, natural ecosystems are characterized by complex and heterogeneous spatial structure. Plants are clustered into patches. Accordingly, herbivores that eat them and the predators that eat the herbivores become similarly arranged in space [1]. This observation was not lost on ecological theorists who in the 1980s and 1990s began to address spatial heterogeneity more explicitly [1],[2]. This new ecological theory, essentially built on additional concepts from physics and chemistry, (e.g., [3]), partitions system dynamics effectively into two phases: a reaction phase in which individuals of species interact locally and a diffusion phase in which individuals disperse after local interactions take place. Dispersal is activated (a positive feedback) in the reaction phase by factors like intense competition or predation risk that causes individuals to move to less competitive or safer locations. Dispersal becomes inhibited (a negative feedback) whenever individuals' efforts to relocate are rebuffed by individuals already occupying the new locations.

This core reaction–diffusion mechanism has been used to develop two distinct classes of theory for species and ecosystem dynamics. The theories differ fundamentally in assumptions about spatial structure and in the way activation and inhibition feedbacks operate on a landscape. One kind of theory (known now as theory of meta-populations and meta-communities) extends classical theory by imposing spatial patch structure as a physical condition of the system (Figure 1) and recasts parameters describing population birth and death processes in terms of spatial movement processes [4]. It then examines the consequences of that structure on system dynamics through analyses of within-patch species interactions and inter-patch species dispersal [4]. The other kind of theory (known now as theory of self-organized systems) starts with a clean slate and examines how spatial structure emerges as a consequence of species interactions and movements [5]. In the meta-system theory, diffusion across a landscape is inhibited by local, within-patch negative feedback (Figure 1). That is, positive and negative feedbacks operate within local patches [4]. In the self-organized systems theory, the positive feedback is local, and negative feedback manifests itself as tension among local clusters of species that prevents further dispersal (Figure 1). Landscape-scale patch structure thus emerges in self-organized systems theory as a consequence of local positive feedback and landscape-scale inhibitive or negative feedback [5].

The upper frames represent the mechanisms determining movement of individuals among discrete patches within a landscape. In the simplest idealization of the theory, individuals within discrete patches belong to a local population. Local populations are connected to each other by individual dispersal (diffusion) that is activated by interactions (reaction) among individuals within a patch. Colonization of another patch by individuals is inhibited whenever dense local populations rebuff dispersing individuals. Dispersal connects the dynamics of local populations to create a grand “meta-population.” The bottom frames represent the mechanics of movement of individuals from a local concentration outward (diffusion), again activated by local interactions among individuals. In the simplest idealization of the theory, full, random dispersion of individuals is inhibited whenever individuals encounter and interact with members of other local concentrations within shared buffer zones. This causes regularly spaced clusters of individuals to become self-organized across the landscape.

Meta-systems theory has gained considerable traction in ecology because it resonates with our intuitive understanding of the current state of many ecosystems [6],[7]. For example, small ponds represent natural, discrete patches within terrestrial landscapes, leading to characteristic patterns of local and landscape-scale species abundances and ecosystem functioning [8]. Human activities have also artificially imposed spatial structure onto many ecosystems by fragmenting formerly continuous landscapes into discrete habitat patches. This has led to predictable transformation of species assemblages and their associated functioning owing to differential abilities of species to reside within patches of a particular size and to disperse among them [9]. Meta-systems theory has clear and profound implications for the conservation of biodiversity [6],[7],[10].

The applicability of self-organized systems theory tends to be less clear because it is a more abstract construct than meta-systems theory. Moreover, there is divided opinion as to whether or not the predicted emergent dynamics based on fairly simple mathematical rules of species engagement are robust to changes in assumptions that reflect real-world ecological conditions [11]. This debate, however, continues to be largely academic because the ultimate arbiter—a rich body of empirical evidence from explicit tests of the theory—has not yet been amassed [5]. There certainly are many putative examples of self-organized, large-scale patterns, owing in good part to advances in satellite imagery [5]. And, there have been efforts to resolve mechanisms driving self-organized pattern formation in species populations [12],[13]. But, evidence that such population-level spatial organization influences whole-ecosystem functioning remains a missing piece of the puzzle.

Testing self-organized systems theory in a whole ecosystem context in nature is not an enterprise for those given to do research yielding quick and simple answers. Unlike meta-systems theory there is no easy and fast way to delineate system structure. Patch boundaries of self-organized systems tend to be fuzzy [5], requiring sophisticated statistical techniques to resolve spatial patterning. The success of this kind of analysis is predicated on obtaining an extensive yet finely resolved data set. Before doing that, however, one must decide what a patch is and what drives the patch structure. For example, does patch structure arise from spatial gradients in soil nutrient concentrations that then cause spatial clumping of plants and the build-up of food chains? Or, does the spatial structure emerge from predator–prey interactions that cause species to emanate away from a local point source? More likely, it is a combination of the two, and so, their relative importance must be resolved through strategic experimentation and sampling of biota and physical conditions. Finally, one must find the points of spatial tension and resolve the mechanisms that delineate patch boundaries. The complexity can be perplexing, leading to the ultimate a priori question: where does one begin?

In this issue of PLoS Biology, Pringle et al. [14] answer these questions while undertaking a herculean effort to explain the spatial patterning of an African savanna ecosystem. Breakthroughs in our understanding of ecological systems often come from having good understanding of the natural history of the system in question and paying attention to the clues that nature provides [15]. Indeed, Pringle et al. [14] capitalize on important prior natural history clues that there is a tendency for termites to exhibit locally non-overlapping foraging territories around their colonies [16]. Termite movement away from the colony seems to be activated by the need to find food, and movement is eventually inhibited when individual termites encounter and compete with members of another colony [16]. Amazingly, this behavior may lead to quite regular spacing of termite colonies across the landscape [14].

More important to the structure and functioning of the entire savanna ecosystem is that the grass-covered mounds created by termite colonies are sandier than surrounding soils. This allows greater water infiltration, aeration, and nutrient build-up on the mounds relative to surrounding soils [14]. Termite mounds are effectively moisture and nutrient “oases” within a dryland matrix. Concentrated moisture and nutrients accordingly promotes tree species growth at the colony margins, with the thickest trees at the immediate mound perimeter and gradually thinner trees emanating away from the perimeter and intergrading with thin trees emanating from other mounds [14]. The nutrient supplied by the mounds to the trees also fosters the build-up of food chains comprised of insect herbivores and spider and lizard predators of the insects.

This emergent structure also leads to parallel activation and inhibition dynamics among species in the food chain [14]. The herbivorous insects are highly concentrated on thicker trees near the mounds and decrease in abundances on thinner, distant trees. Lizards and spiders are likewise dispersed, and field experimentation showed that this was partly because thicker trees offered better hunting sites and partly because prey density was highest on thick trees that tended to be closets to the mounds. A related study [17] shows that nitrogen content of plants is higher near termite mounds, too, meaning that both food quantity and quality is higher near mounds, which likely contributes to all of these patterns.

The combination of nutrient supply for primary plant production, and the translation of plant nutrients into herbivore and predator secondary production mean that termite mounds also become hotspots of ecosystem productivity. These hotspots are preserved through the interplay between activation and inhibition of spatial movement of all of the components of the ecosystem. Thus, the landscape displays a regular pattern of high and low productivity that mirrors the regular patterning of termite mounds. Further statistical modeling suggests that this form of heterogeneity results in greater net productivity than would be expected if the termite mounds were irregularly clustered across the landscape [14]. This derives from the statistical property that when patches are regularly spaced, no single point is very far from a mound, so the productivity of all points when averaged is greater than would be the case when patches are highly clustered or randomly dispersed [14]. Of course, it would be exceedingly difficult to execute the definitive experimental test of this assertion, which would require rearranging the spatial configuration of the termite mounds. This is perhaps the biggest Achilles heel of any empirical effort to test self-organized systems theory within a real-world ecosystem. Nonetheless, the study [14] is exemplary in that it comes the closest yet to satisfying empirical conditions needed to demonstrate the existence of a self-organized ecosystem [5]. By amassing animal behavioral, animal population, and ecosystem data, the authors thus provide a reasonably coherent picture of the spatial mechanisms driving ecosystem structure and functioning.

The intriguing thing is that if, instead of focusing down on a neighborhood of termite mounds, we took a bird's-eye aerial view of the landscape, we could be fooled into concluding that a savannah is a fairly homogeneous landscape. And indeed it is quite plausible to draw such a view of system structure given increased and widespread application of modern satellite imagery to study patterning of savannas and other ecosystems [5],[18],[19]. Then again, if we focused too closely on a termite mound and just its immediate surroundings, our perspective might become so overwhelmed by highly resolved local species interactions that we risk not seeing the spatial patterning at all. The art in empirically resolving the structure and dynamics of self-organized ecosystems is deciding on the appropriate scale of resolution for study [1],[5]. This is not a trivial exercise because it requires years of intrepid field research aimed at understanding both the natural history of a system, and measuring spatial pattern and dynamical processes at many different but complementary spatial perspectives. This may well be the single most important reason why more field evidence for self-organized systems is not yet available.

The study by Pringle et al. [14] nicely shows that theory of self-organized systems is not merely a virtual computer-world phenomenon. There is indeed a basis in “robust reality” [11]. Like meta-systems theory, the implications of self-organized systems theory for conservation, as demonstrated by the study [14], are profound. In this particular case, a very non-charismatic species of fungus-cultivating termite that lives predominantly below-ground seems to create biophysical and biotic conditions that lead to the evolution of aboveground trophic structure and parallel self-organized dynamics in the higher trophic levels. This would, in turn, suggest that the loss of any one of the parts would cause the parallel dynamics sustaining overall ecosystem functioning to quickly collapse. This reinforces the need to consider how the nature of species interactions link to whole-ecosystem functioning when developing strategies to conserve biodiversity [7],[15],[20].


Contents

Resource-resource relationships Edit

Mutualistic relationships can be thought of as a form of "biological barter" [10] in mycorrhizal associations between plant roots and fungi, with the plant providing carbohydrates to the fungus in return for primarily phosphate but also nitrogenous compounds. Other examples include rhizobia bacteria that fix nitrogen for leguminous plants (family Fabaceae) in return for energy-containing carbohydrates. [11]

Service-resource relationships Edit

Service-resource relationships are common. Three important types are pollination, cleaning symbiosis, and zoochory.

In pollination, a plant trades food resources in the form of nectar or pollen for the service of pollen dispersal.

Phagophiles feed (resource) on ectoparasites, thereby providing anti-pest service, as in cleaning symbiosis. Elacatinus and Gobiosoma, genera of gobies, feed on ectoparasites of their clients while cleaning them. [12]

Zoochory is the dispersal of the seeds of plants by animals. This is similar to pollination in that the plant produces food resources (for example, fleshy fruit, overabundance of seeds) for animals that disperse the seeds (service).

Another type is ant protection of aphids, where the aphids trade sugar-rich honeydew (a by-product of their mode of feeding on plant sap) in return for defense against predators such as ladybugs.

Service-service relationships Edit

Strict service-service interactions are very rare, for reasons that are far from clear. [10] One example is the relationship between sea anemones and anemone fish in the family Pomacentridae: the anemones provide the fish with protection from predators (which cannot tolerate the stings of the anemone's tentacles) and the fish defend the anemones against butterflyfish (family Chaetodontidae), which eat anemones. However, in common with many mutualisms, there is more than one aspect to it: in the anemonefish-anemone mutualism, waste ammonia from the fish feeds the symbiotic algae that are found in the anemone's tentacles. [13] [14] Therefore, what appears to be a service-service mutualism in fact has a service-resource component. A second example is that of the relationship between some ants in the genus Pseudomyrmex and trees in the genus Acacia, such as the whistling thorn and bullhorn acacia. The ants nest inside the plant's thorns. In exchange for shelter, the ants protect acacias from attack by herbivores (which they frequently eat, introducing a resource component to this service-service relationship) and competition from other plants by trimming back vegetation that would shade the acacia. In addition, another service-resource component is present, as the ants regularly feed on lipid-rich food-bodies called Beltian bodies that are on the Acacia plant. [15]

In the neotropics, the ant Myrmelachista schumanni makes its nest in special cavities in Duroia hirsute. Plants in the vicinity that belong to other species are killed with formic acid. This selective gardening can be so aggressive that small areas of the rainforest are dominated by Duroia hirsute. These peculiar patches are known by local people as "devil's gardens". [16]

In some of these relationships, the cost of the ant's protection can be quite expensive. Cordia sp. trees in the Amazonian rainforest have a kind of partnership with Allomerus sp. ants, which make their nests in modified leaves. To increase the amount of living space available, the ants will destroy the tree's flower buds. The flowers die and leaves develop instead, providing the ants with more dwellings. Another type of Allomerus sp. ant lives with the Hirtella sp. tree in the same forests, but in this relationship, the tree has turned the tables on the ants. When the tree is ready to produce flowers, the ant abodes on certain branches begin to wither and shrink, forcing the occupants to flee, leaving the tree's flowers to develop free from ant attack. [16]

The term "species group" can be used to describe the manner in which individual organisms group together. In this non-taxonomic context one can refer to "same-species groups" and "mixed-species groups." While same-species groups are the norm, examples of mixed-species groups abound. For example, zebra (Equus burchelli) and wildebeest (Connochaetes taurinus) can remain in association during periods of long distance migration across the Serengeti as a strategy for thwarting predators. Cercopithecus mitis and Cercopithecus ascanius, species of monkey in the Kakamega Forest of Kenya, can stay in close proximity and travel along exactly the same routes through the forest for periods of up to 12 hours. These mixed-species groups cannot be explained by the coincidence of sharing the same habitat. Rather, they are created by the active behavioural choice of at least one of the species in question. [17]

Mathematical treatments of mutualisms, like the study of mutualisms in general, has lagged behind those of predation, or predator-prey, consumer-resource, interactions. In models of mutualisms, the terms "type I" and "type II" functional responses refer to the linear and saturating relationships, respectively, between benefit provided to an individual of species 1 (y-axis) on the density of species 2 (x-axis).

Type I functional response Edit

One of the simplest frameworks for modeling species interactions is the Lotka–Volterra equations. [18] In this model, the change in population density of the two mutualists is quantified as:

Mutualism is in essence the logistic growth equation + mutualistic interaction. The mutualistic interaction term represents the increase in population growth of species one as a result of the presence of greater numbers of species two, and vice versa. As the mutualistic term is always positive, it may lead to unrealistic unbounded growth as it happens with the simple model. [19] So, it is important to include a saturation mechanism to avoid the problem.

Type II functional response Edit

In 1989, David Hamilton Wright modified the Lotka–Volterra equations by adding a new term, βM/K, to represent a mutualistic relationship. [20] Wright also considered the concept of saturation, which means that with higher densities, there are decreasing benefits of further increases of the mutualist population. Without saturation, species' densities would increase indefinitely. Because that isn't possible due to environmental constraints and carrying capacity, a model that includes saturation would be more accurate. Wright's mathematical theory is based on the premise of a simple two-species mutualism model in which the benefits of mutualism become saturated due to limits posed by handling time. Wright defines handling time as the time needed to process a food item, from the initial interaction to the start of a search for new food items and assumes that processing of food and searching for food are mutually exclusive. Mutualists that display foraging behavior are exposed to the restrictions on handling time. Mutualism can be associated with symbiosis.

Handling time interactions In 1959, C. S. Holling performed his classic disc experiment that assumed the following: that (1), the number of food items captured is proportional to the allotted searching time and (2), that there is a variable of handling time that exists separately from the notion of search time. He then developed an equation for the Type II functional response, which showed that the feeding rate is equivalent to

The equation that incorporates Type II functional response and mutualism is:

  • N and M=densities of the two mutualists
  • r=intrinsic rate of increase of N
  • c=coefficient measuring negative intraspecific interaction. This is equivalent to inverse of the carrying capacity, 1/K, of N, in the logistic equation.
  • a=instantaneous discovery rate
  • b=coefficient converting encounters with M to new units of N

This model is most effectively applied to free-living species that encounter a number of individuals of the mutualist part in the course of their existences. Wright notes that models of biological mutualism tend to be similar qualitatively, in that the featured isoclines generally have a positive decreasing slope, and by and large similar isocline diagrams. Mutualistic interactions are best visualized as positively sloped isoclines, which can be explained by the fact that the saturation of benefits accorded to mutualism or restrictions posed by outside factors contribute to a decreasing slope.

The type II functional response is visualized as the graph of b a M 1 + a T H M <1+aT_M>>> vs. M.

Mutualistic networks made up out of the interaction between plants and pollinators were found to have a similar structure in very different ecosystems on different continents, consisting of entirely different species. [21] The structure of these mutualistic networks may have large consequences for the way in which pollinator communities respond to increasingly harsh conditions and on the community carrying capacity. [22]

Mathematical models that examine the consequences of this network structure for the stability of pollinator communities suggest that the specific way in which plant-pollinator networks are organized minimizes competition between pollinators, [23] reduce the spread of indirect effects and thus enhance ecosystem stability [24] and may even lead to strong indirect facilitation between pollinators when conditions are harsh. [25] This means that pollinator species together can survive under harsh conditions. But it also means that pollinator species collapse simultaneously when conditions pass a critical point. [26] This simultaneous collapse occurs, because pollinator species depend on each other when surviving under difficult conditions. [25]

Such a community-wide collapse, involving many pollinator species, can occur suddenly when increasingly harsh conditions pass a critical point and recovery from such a collapse might not be easy. The improvement in conditions needed for pollinators to recover could be substantially larger than the improvement needed to return to conditions at which the pollinator community collapsed. [25]

Humans are involved in mutualisms with other species: their gut flora is essential for efficient digestion. [27] Infestations of head lice might have been beneficial for humans by fostering an immune response that helps to reduce the threat of body louse borne lethal diseases. [28]

Some relationships between humans and domesticated animals and plants are to different degrees mutualistic. For example, agricultural varieties of maize provide food for humans and are unable to reproduce without human intervention because the leafy sheath does not fall open, and the seedhead (the "corn on the cob") does not shatter to scatter the seeds naturally. [29]

In traditional agriculture, some plants have mutualist as companion plants, providing each other with shelter, soil fertility and/or natural pest control. For example, beans may grow up cornstalks as a trellis, while fixing nitrogen in the soil for the corn, a phenomenon that is used in Three Sisters farming. [30]

One researcher has proposed that the key advantage Homo sapiens had over Neanderthals in competing over similar habitats was the former's mutualism with dogs. [31]

Mutualism breakdown Edit

Mutualisms are not static, and can be lost by evolution. [32] Sachs and Simms (2006) suggest that this can occur via 4 main pathways:

  1. One mutualist shifts to parasitism, and no longer benefits its partner, [32] such as headlice [citation needed]
  2. One partner abandons the mutualism and lives autonomously [32]
  3. One partner may go extinct [32]
  4. A partner may be switched to another species [33]

There are many examples of mutualism breakdown. For example, plant lineages inhabiting nutrient-rich environments have evolutionarily abandoned mycorrhizal mutualisms many times independently. [34]


Biological Interactions: Positive and Negative Interactions in an Ecosystem (.PPT)

Organisms living together in a community influence each other directly or indirectly under natural conditions. All the vital process of living such as growth, nutrition and reproduction requires such interactions between individuals in the same species (intraspecific) or between species (interspecific) These inter or intra relationships of individuals in a population or community of an ecosystem is called biological interactions or population interactions.

The interaction between organisms may not be always beneficial to all the interacting counter parts. Based on whether, the interaction is beneficial to both interacting species or harmful to at least one interaction species, the ecological of biological interactions are classified into two categories.

(I). Positive interactions

(II). Negative interactions

(I). Positive interactions:

In positive interactions, the interacting populations help one another. The positive interaction may be in one way or reciprocal. The benefit may be in respect of food, shelter, substratum or transportation. The positive association may be continuous, transitory, obligate or facultative. The two interacting partners may be in close contact in such a way that the tissues intermixed with each other or they may live within a specific area of the other or attached to its surface. Different types of positive population interactions are:

(1). Mutualism

(2). Commensalism

(3). Proto-cooperation

(1). Mutualism:

Mutualism, also called as symbiosis, is also a positive type of ecological interaction. Mutualism is a symbiotic association between two organisms in which both the interacting partners are mutually benefitted. Mutualism is different from proto-cooperation in the sense that mutualism is obligatory and none of the partners of mutualism can survive individually. In mutualism, the organisms enter into some sort of physical and physiological exchange.

Lichen (symbiotic association between algae and fungi)

(a). Lichens: lichens are the symbiotic association between algae and fungi. The body of lichen composed of fungal matrix in which the algal cells are embedded. The fungi provide protection to algal components and also provide moisture and nutrients to them. The algal components in turn will supply carbohydrates for fungus.

(b). Symbiotic nitrogen fixation: mutualistic interaction can be seen in the symbiotic nitrogen fixation of Rhizobium associated with root nodules of leguminous plants is the best example. Similarly other microorganisms associated with plants such as Alnus, Casuarina, Cycas for nitrogen fixation are also belongs to mutualism.

(c). Mycorrhizae: they are the symbiotic association between fungi and the roots of some trees. Fungal components help in the absorption of water and minerals by the plant. The plant in turn supplies foot to fungal components.

(d). Pollination by animals: Bees, moths, butterflies etc. derive food from the nectar of plants and in return bring out pollination

(e). Seed dispersal by animals: Fruits are eaten by birds, and other animals and the seeds contained in them are dropped in the excrement at various places.

(f). Zoochlorellae and Zooxanthellae: Zoochlorellae and Zooxanthellae are unicellular microscopic algae that symbiotically live in the outer tissue of some sponges, coelenterates and mollusks. Algae are autotrophs and they can prepare food by photosynthesis. Algae obtain materials released by metabolism of host animals for their photosynthesis. Chlorella vulgaris is a unicellular green alga which lives in the gastro-dermal cells of Hydra. Algae through photosynthesis provide food and oxygen to Hydra, which in turn provide shelter, nitrogen wastes and CO2 to Chlorella.

(g). Association between termites and Trichonympha: Termites feeds on wood, however they cannot digest the cellulose in the wood. Trichonympha is a protozon which lives in the gut of termites. Trichonympha can produce digestive enzymes and they digest cellulose of wood. Trichonympha in turn obtain food and shelter from termite.

(2). Commensalism:

Commensalism is a positive type of ecological interaction between two species in an ecosystem. In commensalism, the association occurs between members of two different species where one species is benefited the other is neither benefited nor harmed. Here the two populations live together without entering into any kind of physical exchange, and one is benefited without any effect on the other.

(a). Climbers and lianas such as Bauhinia, Tinospora etc., which are rooted in the soil but climb over large trees. These climbers use other trees as support to get enough sunlight, more than that, the supporting plants do not have any positive or negative effect.

(b). Epiphytes: They are the plants which growing on the surface of other large plants. They use other plants only as a support and not for water or food supply. They are different from lianas in that they are not rooted in the soil. Example: Orchids, Mosses, Nephrolepis, Usnea, green algae growing on the surface of snails, microbes such as bacteria and protozoans live within the body cavity of other animals.

(3). Proto-cooperation:

Proto-cooperation is a positive type of population interaction and it is also called as non-obligatory mutualism. Proto-cooperation is a less extreme type of population interaction. In proto-cooperation, two species interact favourably with each other, though both of them are able to survive separately. It is a temporary association where both the interacting partners get benefited. It is different from mutualism in the sense that, the association is not essential for the survival of any of the species.

Example for proto-cooperation: Association between hermit crab (Eupagurus prideauxi) and sea anemone. The sea anemone is carried by the carb to fresh feeding sites and the crab is in turn protected from enemies by sea anemone.

(II). Negative interactions:

In negative interactions, one of the interacting populations is benefited and the other is harmed. In negative interaction one population may eat members of the other population, compete for foods or excrete harmful wasters. Different types of negative population interactions are:

(1). Ammensalism

(2). Parasitism

(3). Predation

(4). Cannibalism

(5). Competition

(1). Ammensalism:

Ammensalism is a negative type of population interaction. In ammensalism one species is harmed or inhibited other is neither benefitted nor harmed. Some authors prefer to use the term antibiosis for commensalism. Antibiosis is the partial or complete inhibition or death of one organism by another through the production of some substances or environmental conditions as a result of its metabolic pathway. In antibiosis none of them derives any benefit. The process of antibiosis is common in microbial populations and the chemical substances produced by microbes for antibiosis are generally called as antibiotics.

(a). Chlorella vulgaris produces a toxin (chlorellin, an antibiotic) which is harmful to other algae.

(b). Larger and more powerful organism excludes another organism from its source of shelter or food is also a type of ammensalism

(c). Algal blooms such as red tide or green blooms are also example of ammensalism.

(2). Parasitism:

Parasitism is a negative type of population interaction. Parasitism belongs to the ‘exploitation’ category of negative population interactions. In exploitation, one species harms the other by making its direct or indirect use for shelter or food. A parasite is the organism living on or in the body of another organisms and deriving food form its tissues. The harmed one is called host, the benefitted one is called parasite. A parasite usually takes a host which is usually larger than its body size. Usually a specialized parasite does not kill the host at least until it has completed its reproductive cycle. Those organisms which derive their nourishment only partly and remain in contact with their host only for a short period of their life cycle are not true parasites (examples: mosquitos). Some parasites requires more than one host to complete its life cycle and such parasites are called heteraceous parasites (example Puccinia, Malarial parasite).

(a). Cuscuta is a total stem parasite which lives on the surface of other large plants. They are devoid of chloroplasts and hence they cannot prepare their own food. Thy have specialized absorptive structures called haustoria. In the case of complete parasite, the haustoria will be inserted into the phloem tissue of host plants and they absorb the prepared food materials from the host phloem.

(b). Rafflesia, Orabanche and Conopholis are complete root parasites

(c). Loranthus and Viscum (Loranthaceae) are partial stem parasites. They bear leaves with chlorophylls and hence they can prepare their own food. The haustoria of partial parasite are attached to the xylem of host plants. Form the xylem of host plants, partial parasite absorbs water and minerals and they prepare their own food by photosynthesis.

(d). Santalum album and Thesium are partial root parasites. Their roots are attached to the host plants.

(e). Microorganisms such as bacteria, virus, fungi, mycoplasma, protozoans etc. which cause many diseases in human and other animals and plants are parasites.

(f). Hyperparasites: Parasitic microbes growing in or on other parasites are called hyper parasites.

(3). Predation:

Predation is a negative type of population interaction and it belongs to the ‘exploitation’ category of negative population interactions. In predation, one species kill and feeds on another species. The killer species is called predator and the one who dead are called prey. The predators are usually larger and power-full than prey. Predation is very important in community dynamics and it helps to maintain the constancy of number of different trophic levels in the ecosystem and thereby maintain the stability of ecosystem.

(a). Lion, tiger and Beer are predators of forest ecosystem. They predate herbivores

(4). Cannibalism

Cannibalism is a negative type of interaction of individuals in the same population. In cannibalism, bigger individual of a species kill and feeds on smaller individual of same species. Cannibalism is a natural method of population control in the ecosystem.

(5). Competition

Competition is the association of two or more species each species is adversely affected by the presence of other species in respect of food, shelter, space, light etc. Competition occurs when individuals attempt to obtain a resource that is inadequate to support all the individuals seeking it or even if the resources are adequate individuals harm one another in trying to obtain it. The resources in the environment for which the individuals compete include raw materials for life such as water, light and nutrients, space for occupying and selection of mates for sexual reproduction. The competition in the ecosystem may be of two types:

a. Intraspecific competition

b. Interspecific competition

(a). Intra-specific competition: It is the competition occurring between the individuals of the same population (competition within population). It is also called as scramble competition. Intra-specific competition is an important density dependent factor regulating population size. Intra-specific competition is also responsible for the even distribution of individuals of the species in an ecosystem.

(b). Inter-specific competition: It is the competition occurring between populations of different species whose requirements are common and inadequate in the ecosystem (competition between population). It is also called as contest or interference competition.

Key concepts:

What are biological / ecological / population interactions?
What are the different types of population interaction in the ecosystem?
What is meant by mutualism (symbiosis)?
What is meant by commensalism?
What is proto-cooperation?
What is ammensalism?
What is parasitism?
What is predation?
What is cannibalism?
What is meant by competition in an ecosystem?
Differentiate intraspecific and interspecific competition


How Do Species Interactions Affect Evolutionary Dynamics Across Whole Communities?

Theories of how species evolve in changing environments mostly consider single species in isolation or pairs of interacting species. Yet all organisms live in diverse communities containing many hundreds of species. This review discusses how species interactions influence the evolution of constituent species across whole communities. When species interactions are weak or inconsistent, evolutionary dynamics should be predictable by factors identified by single-species theory. Stronger species interactions, however, can alter evolutionary outcomes and either dampen or promote evolution of constituent species depending on the number of species and the distribution of interaction strengths across the interaction network. Genetic interactions, such as horizontal gene transfer, might also affect evolutionary outcomes. These evolutionary mechanisms in turn affect whole-community properties, such as the level of ecosystem functioning. Successful management of both ecosystems and focal species requires new understanding of evolutionary interactions across whole communities.


Setting Realistic Recovery Targets for Two Interacting Endangered Species, Sea Otter and Northern Abalone

Failure to account for interactions between endangered species may lead to unexpected population dynamics, inefficient management strategies, waste of scarce resources, and, at worst, increased extinction risk. The importance of species interactions is undisputed, yet recovery targets generally do not account for such interactions. This shortcoming is a consequence of species-centered legislation, but also of uncertainty surrounding the dynamics of species interactions and the complexity of modeling such interactions. The northern sea otter (Enhydra lutris kenyoni) and one of its preferred prey, northern abalone (Haliotis kamtschatkana), are endangered species for which recovery strategies have been developed without consideration of their strong predator–prey interactions. Using simulation-based optimization procedures from artificial intelligence, namely reinforcement learning and stochastic dynamic programming, we combined sea otter and northern abalone population models with functional-response models and examined how different management actions affect population dynamics and the likelihood of achieving recovery targets for each species through time. Recovery targets for these interacting species were difficult to achieve simultaneously in the absence of management. Although sea otters were predicted to recover, achieving abalone recovery targets failed even when threats to abalone such as predation and poaching were reduced. A management strategy entailing a 50% reduction in the poaching of northern abalone was a minimum requirement to reach short-term recovery goals for northern abalone when sea otters were present. Removing sea otters had a marginally positive effect on the abalone population but only when we assumed a functional response with strong predation pressure. Our optimization method could be applied more generally to any interacting threatened or invasive species for which there are multiple conservation objectives.

Definición de Metas de Recuperación Realistas para Dos Especies en Peligro Interactuantes, Enhydra lutris y Haliotis kamtschatkana

Resumen

La falta de considerar las interacciones entre especies en peligro puede llevar a dinámicas poblacionales inesperadas, estrategias de manejo ineficientes, despilfarro de recursos escasos, y, peor aun, incremento en el riesgo de extinción. La importancia de las interacciones de especies no está en disputa, sin embargo los objetivos de recuperación generalmente no toman en cuenta a dichas interacciones. Este problema es una consecuencia de la legislación centrada en especies, pero también de la incertidumbre que rodea a dinámica de las interacciones de especies y la complejidad para modelar esas interacciones. La nutria marina (Enhydra lutris kenyoni) y una de sus presas preferidas, el abulón (Haliotis kamtschatkana), son especies en peligro para las que se han desarrollado estrategias de recuperación sin considerar sus estrechas interacciones depredador-presa. Utilizando procedimientos de optimización basados en simulaciones de inteligencia artificial, específicamente aprendizaje por reforzamiento y programación dinámica estocástica, combinamos modelos poblacionales de nutria y abulón con modelos de respuesta funcional y examinamos como afectan diferentes acciones de manejo a la dinámica de la población y la probabilidad de alcanzar las metas de recuperación de cada especie en el tiempo. Fue difícil alcanzar las metas de recuperación para estas especies interactuantes en la ausencia de manejo. Aunque se pronosticó que la población de nutrias se recuperaría, el logro de las metas de recuperación de abulón falló no obstante que las amenazas, como la depredación y captura furtiva, fueron reducidas. Una estrategia de manejo que conlleve una reducción de 50% en la captura furtiva de abulón fue el requisito mínimo para alcanzar metas de recuperación de abulón a corto plazo cuando había presencia de nutrias. La remoción de nutrias tuvo un efecto positivo marginal sobre la población de abulón pero solo cuando asumimos una respuesta funcional con fuerte presión de depredación. Nuestro método de optimización pudiera ser aplicado más generalmente a cualquier especie amenazada o invasora interactuante para la cual se hayan fijado múltiples objetivos de conservación.


Teaching Focus

As a tenure-track assistant professor at GMU, I engage students in environmental science through innovative classes incorporating field work and hands-on activities, a research program focusing on aquatic invertebrate ecology and biology with many opportunities for student involvement, and contribute to the community through outreach education and collaboration with local, state and national environmental decision makers.

EVPP/BIOL 350: Freshwater Ecosystems Lecture and Lab (UG)
EVPP/BIOL 449: Marine Ecology (G, UG)
EVPP 301: Environmental Science: Biological Diversity and Ecosystems (UG)
EVPP 490: Aquatic Invertebrate Ecology Lecture and Lab (G, UG)
EVPP 991: Experimental Design for Environmental Scientists (G)
EVPP 505: Experimental Design and Statistics for Environmental Science (G)


Box 1. The Interaction Compass

Interactions are usually defined by the direction in which they affect the interactors, be they species, strains, or individuals. Even as variation in interspecific interactions first came into focus [1,9], it was clear that both the strength and the sign of interactions shifted back and forth along a continuum (Fig 1). The center of the interaction compass (see [10,11]) is sometimes called neutralism, but this box classifies any interaction where a fitness effect does not occur. Although the interaction compass is typically shown with only two species for the purposes of illustration, all species are involved in networks of interactions, and indirect interactions—defined where one species affects another by way of a third species or pathway—are ubiquitous in ecological communities and can rival direct interactions in their strength [e.g., 12]. The variety of terms and their distinct historical origins can lead to some ambiguity, as is the case with mutualism and facilitation [13]. Facilitation does not appear in the interaction compass (Fig 1), but it is associated with some of the earliest research on positive interactions across environmental gradients and with the stress gradient hypothesis in particular [4]. The term arises from 20th-century plant community ecology and refers either to any interaction where one species modifies the environment in a way that is positive for a neighboring species or specifically to positive interactions within a trophic level. Relevant here, until the recent surge of interest in microbe-microbe interactions, the term mutualism typically referred to interactions between trophic levels, where the competition outcome (––) is unlikely because the interactors do not overlap substantially in their niche requirements. It is common to speak instead of the mutualism-parasitism continuum. Although microbes fit perhaps only uncomfortably into the trophic boxes defined on the basis of macroorganism interactions, most cross-feeding mutualisms occur within a trophic level and thus could be thought of as examples of both mutualism and facilitation, with outcomes ranging around the full compass, from mutualism to competition and back to mutualism again.

A two-species interaction is illustrated with the terms defining each of the differently signed outcomes the signs indicate individual fitness or population growth rate. A positive (+) sign thus indicates a positive effect of the interaction on the individual or population, a zero (0) sign indicates no effect, and a negative (–) sign indicates a negative effect. Moving away from the center increases the magnitude of the net effect of the interaction.

One relatively straightforward path by which an increase in stress can lead to stronger mutualism is when the interaction involves a direct exchange of the environmentally limiting resource. In North American grasslands, grasses associate with arbuscular mycorrhizal fungi, which exchange soil nutrients for carbon fixed by the grass. The fungi can deliver both phosphorus and nitrogen, increasing grass uptake of whichever nutrient is least available in a given soil [5]. Although this seems like a good trick, we cannot characterize the outcome of the grass's interaction with the fungi on the basis of nutrient uptake (the benefit) alone. The delivery of carbon by the grass to the fungi (the cost), and the net balance of trade (benefit−cost), is key. In this example, the grass receives a net benefit (increased biomass) from interacting with the fungi in phosphorus-poor soil but not in nitrogen-poor soil [5]. The fungi thus seem to be parasites in nitrogen-poor soil, but, interestingly, even that interaction is less negative for grasses in soils with less nitrogen [5]. This example suggests potentially broad relevance for the stress gradient hypothesis across the continuum of interaction types (Box 1) (Fig 1) but also highlights how the balance of trade determines ecological outcomes. To predict the outcome of any given interaction, therefore, we need to understand how both the benefits and the costs of interactions depend on an organism's environment. This is a challenging task, requiring integrative understanding of organismal physiology, axes of environmental variation, and the nature of biotic interactions, as well as of the feedbacks between organismal ecology and evolution.

The diversity and experimental tractability of microorganisms, as well as their fundamental role in life on earth, make them appealing systems for studying context dependence in its multiple dimensions. The potential for mutualism among microbes and between microbes and their multicellular hosts is receiving unprecedented attention as the diverse and important roles of the human gut microbiome come into sharp relief. As field-based studies of macroorganism interactions move past the recognition of context dependence to a deliberate focus on its drivers and mechanisms [14], laboratory-based studies of microbes are, in parallel, moving past debates about the "typical" nature of microbial interactions [15,16] to focus on how and why interaction outcomes vary across environmental gradients [3,17].

In this issue of PLOS Biology, Hoek and colleagues show that interactions between two cross-feeding yeast strains can transition across nearly a full continuum of outcomes with simple variation in environmental nutrient concentration [18]. Cross-feeding microbes are those with similar metabolic requirements whose metabolic pathways are complementary, either because of a "leaky" byproduct system whereby some metabolites end up in the environment [15] or because of costly, cooperative exchange [19]. In the Hoek et al. study, the investigators used strains of cross-feeding yeast engineered to differ in amino acid production: one strain lacks leucine production but overproduces tryptophan (Leu − ), and the other lacks tryptophan production but overproduces leucine (Trp − ). By varying the quantity of leucine and tryptophan in the environment in a constant ratio, the investigators produced a continuum of interaction outcomes, from low-amino-acid environments that exhibit obligate mutualism to high-amino-acid environments that exhibit strong competition. They go on to show that many of these dynamics can be recovered with a remarkably simple model of each strain's population growth, primarily depending only on the quantity of environmental amino acids and the population densities of the two strains. The complete range of empirically determined qualitative change in the interaction is mirrored by this simple model, which suggests that the outcomes of interactions that depend on resource exchange (including most mutualisms! [20]) can be predicted to an impressive degree by measuring the availability of that resource, the population densities of the interacting species, and their intrinsic growth rates.

Returning to our grass-fungi interaction from above, however, we recall that the benefits of interacting (receiving the missing amino acid in the case of these yeasts) are only one side of the coin. The costs of interacting are what underlie the conflicts of interest that threaten mutualism stability and can lead to increasingly negative interactions over evolutionary time. In the Hoek et al. study, the costs of overproducing the amino acid that is consumed by the other strain are modeled only implicitly in the intrinsic growth rate (r), which is determined by growing each strain in monoculture with unlimited amino acids. The major discrepancy between their model and the empirical data, however, is that the model incorrectly predicts a much larger range of amino-acid concentrations at which the Trp − strain is expected to outperform the Leu − strain in both monoculture and co-culture. Interestingly, the Trp − strain performs particularly poorly when it is co-cultured with the Leu − strain, except at very high levels of environmental amino-acid availability. It is tempting to speculate that this discrepancy is caused by the model's lack of an explicit density-dependent cost of leucine production, which would be exacerbated when the Leu − strain is performing well. Going forward, it should be possible to merge population dynamic models that include such a cost [21] with an explicit term for resource availability to see how well these models predict dynamics in a variety of empirical systems.

Rapid, ongoing global change presents one compelling reason to determine how resource availability underpins interactions, and the Hoek et al. study also sheds some needed light here. Using their model, they determine distinct early-warning "signatures" of imminent population collapse for co-cultures at very low levels of amino acids (think, e.g., drought), in which both strains go extinct upon the collapse of the obligate mutualism, and at very high levels of amino acids (think, e.g., nutrient pollution), in which competitive exclusion leads to the extinction of the slower growing strain. The collapse of populations engaging in obligate mutualism is predicted when the ratio of the population densities of the two strains becomes stable much more quickly than the total population size (particularly at small population sizes) and vice versa for competitive exclusion. In contrast, in healthy populations, the ratio of the strains and the total population size become stable at approximately equal speeds. As the authors note, this result suggests that we can predict how close one or both interacting species are to extinction by monitoring their comparative population dynamics.

This is an exciting prospect, but how easy is it to monitor the population dynamics of interacting species outside the laboratory? Monitoring populations of long-lived species in the field is inherently difficult, and, historically, less attention has been paid to determining how the environment affects populations of interacting species than to how it affects individual traits and fitness [14]. But wait, you say, surely individual fitness is the driving force behind population dynamics. Well, yes and no. To assess individual fitness, investigators almost always use one or more proxies, including growth, survival, and reproductive biomass. Population-level studies have shown that a given interacting species can have multiple, frequently opposing effects on these different components of their partner's fitness, such that an exclusive focus on any one component can be very misleading [22,23]. In addition, population dynamics depend on the probability of successful offspring recruitment this probability is critical because it itself is also likely to vary along environmental gradients [14]. Population-level approaches thus deserve explicit focus despite their challenges, and this is one area where field studies can be greatly enhanced by both theory and model systems.


Mammalian Nkx2.2+ perineurial glia are essential for motor nerve development.
Kucenas S
Developmental dynamics : an official publication of the American Association of Anatomists 243.9 (2014 Sep): 1116-29.

Sox2 activates cell proliferation and differentiation in the respiratory epithelium.
Whitsett JA
American journal of respiratory cell and molecular biology 45.1 (2011 Jul): 101-10.

Characterization of an antigenic determinant preferentially expressed by type I epithelial cells in the murine thymus.
Hosier S
The journal of histochemistry and cytochemistry : official journal of the Histochemistry Society 40.5 (1992 May): 651-64.

Characterization of an antigenic determinant preferentially expressed by type I epithelial cells in the murine thymus.
Hosier S
The journal of histochemistry and cytochemistry : official journal of the Histochemistry Society 40.5 (1992 May): 651-64.

Mammalian Nkx2.2+ perineurial glia are essential for motor nerve development.
Kucenas S
Developmental dynamics : an official publication of the American Association of Anatomists 243.9 (2014 Sep): 1116-29.

Characterization of an antigenic determinant preferentially expressed by type I epithelial cells in the murine thymus.
Hosier S
The journal of histochemistry and cytochemistry : official journal of the Histochemistry Society 40.5 (1992 May): 651-64.

Novel role for ALCAM in lymphatic network formation and function.
Halin C
FASEB journal : official publication of the Federation of American Societies for Experimental Biology 27.3 (2013 Mar): 978-90.

Characterization of an antigenic determinant preferentially expressed by type I epithelial cells in the murine thymus.
Hosier S
The journal of histochemistry and cytochemistry : official journal of the Histochemistry Society 40.5 (1992 May): 651-64.

Mammalian Nkx2.2+ perineurial glia are essential for motor nerve development.
Kucenas S
Developmental dynamics : an official publication of the American Association of Anatomists 243.9 (2014 Sep): 1116-29.


8.1: Dynamics of two Interacting Species - Biology

MATH 304 - MATHEMATICAL MODELING: DETERMINISTIC MODELS

Required : Edelstein-Keshet, L. (2005) Mathematical Models in Biology . SIAM
Optional: Higham, D.J. and Higham, N.J. (2005) MATLAB Guide, Second Edition. SIAM
or Driscoll, T.A. (2009) Learning MATLAB. SIAM
(Recommended if you have never used Matlab.)

Office hours: Tuesday 12/15 1-2 pm Wednesday 12/16 10-11 am in Keller 409 or by appointment.
Review session: Tuesday 12/15 3:30 - 5:30 pm in the zoology department seminar room, 152 Edmondson Hall.

  • Syllabus
  • Academic Expectations
  • Exam 1 solutions
  • Project
  • Exam 2 solutions
  • M ATLAB Intro
  • Lab session 1
  • Lab session 2
  • MATLAB Basics
  • Plotting commands
  • dlogistic.m
  • diter.m
  • dlogplot.m
  • diter2.m
  • Lab session 3
  • Lab session 7
  • myf.m
  • lotkav.m
  • predator.m
  • vdp1.m
  • Lab session 9
  • Lab session 11
  • HW #1: C hapter 1, #2d,f * , 10 Chapter 2, #1a,b,c,e, 2b,c, 3
  • HW #2: Chapter 2, #4a,b,c, 9, 10 (0<b<1,b=1,b>1), 14, Chapter 3, #1
  • Lab exercise 1
  • HW #3: Chapter 3, #5 Chapter 4, #3, 4, 5a,d,f, 15
  • Lab exercise 2
  • HW #4: Chapter 4, #7a,b, 9, 10a,b, 14a,b, 17
  • Lab exercise 3
  • HW #5: Chapter 4, 16a,b,c, 22a,b, 25a-d
  • Lab exercise 4
  • HW #6: Chapter 5, #5a,c,e,f, 6a,c,e,f
  • HW #7: Chapter 5, #7c,d,e,f, 11
  • HW #8: Chapter 6, #3, 8, 17(equations in Errata pg. xxxvii)
  • HW #9 : Chapter 8, #2, 4a,b, 7a,c,d, 10, 16a,c
  • HW #1 solutions
  • HW #2 solutions
  • HW #3 solutions
  • HW #4 solutions
  • HW #5 solutions
  • HW #6 solutions
  • HW #7 solutions
  • HW #8 solutions
  • HW #9 solutions

Week 1 Introduction: Why Model? discrete time models for population dynamics, linear difference equations (Chapter 1)

Week 2 Introduction to nonlinear discrete dynamical systems: graphical analysis, fixed points, linear stability analysis, bifurcation, chaotic dynamics, systems of difference equations (Chapter 2)

Week 3 Applications of nonlinear difference equations density-dependent population models (Chapter 3)

Week 4 Introduction to continuous time models: logistic equation for single species population dynamics (Chapter 4)

Week 5 Some techniques for ordinary differential equations: equilibrium points, stability, linearization (Chapter 4)

Week 6 Introduction to continuous dynamical systems: geometric (phase plane) analysis of 2-dim systems, linear systems (Chapter 5) Exam 1

Week 7 Nonlinear systems, periodic solutions (Chapter 5)

Week 8 Continuous time models for single species population dynamics: harvesting, metapopulations (patchy environments)

Week 9 Interacting populations: predator and prey models, competition, infectious disease models (Chapter 6)

Week 10 Biochemical kinetics (Chapter 7)

Week 11 Hodgkin-Huxley model for nerve conduction, Exam 2

Week 12 Poincaré-Bendixon theory, oscillations in simplified nerve models (FitzHugh-Nagumo equations) (Chapter 8)

Week 13 Hopf bifurcations (Chapter 8) cable equation

Week 14 Introduction to partial differential equations (Chapter 9)

Week 15 Introduction to reaction-diffusion equations, traveling wave solutions (Chapter 10) Project due


Watch the video: Valení a síly. 910 Dynamika. Fyzika. (August 2022).