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Living organisms exist within webs of interactions with other living creatures, the most important of which involve eating or being eaten (trophic interactions). Fig. 1. A food chain consisting of three trophic levels, plant (P), herbivore (H) and carnivore (C), and their interactions andfeedbacks.

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There are three basic ways in which the modeling of trophic dynamics has been approached Consider population interactions and built the models around the law of "mass action". Considering the energy-flow approach by modeling the general process of consumption, or the flow of biomass through the food chain. Modeling the effects of plant and carnivore densities on the reproduction and survival of the herbivore population Lotka-Volterra Model The mass action approach to modeling trophic interactions was pioneered, independently, by the American physical chemist Lotka (1925) and Italian mathematician Volterra (1926).

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These authors argued that consumer and resource populations could be treated like particles interacting in a homogeneously mixed gas or liquid and, under these conditions, the rate of encounter between consumers and resources (the reaction rate) would be proportional to the product of their masses aH = the growth rate of the herbivore population in the absence of predators. Thus, in the absence of predators, the herbivore population grows according to the equation dH/dt = aH, which integrates to Ht = H0 Exp(at) bHC = the rate of consumption of herbivores, or their death rate due to attack by predators. cbHC = the rate of production of predator offspring, which is directly related to the number of prey consumed. mC = the death rate of consumers in the absence of food. Thus, in the absence of prey, predators die according to dC/dt = -mC, which integrates to Ct = C0Exp(-mt) The prey and predator populations will be in equilibrium when their per- capita rates of change are zero (per- capita -> (dA/dt).(1/A) called R functions ). The equilibrium isoclines are : C = a/b and H = m/cb

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Figure. (Left) Herbivore (H) and carnivore (C) phase space showing zero-growth isoclines for the L-V model (herbivore isocline = horizontal blue line, carnivore isocline = vertical red line). Arrows show directional vectors of population change in the four regions of phase space. (Right) Herbivore (blue) and carnivore (red) time series plot showing cycles of abundance with the predator cycles lagging behind the prey. Conditions for plausible predator-prey models Royama (1992) and Berryman et al. (1995) proposed the following minimal set of attributes that all consumer-resource models should possess: Attribute1: The death rate of an individual in the resource population should increase with the density of the consumer population; i.e., the more predators there are, the greater your probability of being eaten.

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Attribute 2: The birth rate of individual consumers should increase with the density of the resource population,the more food one gets, the more offspring one produces. Attribute 3: The reproductive rate of a consumer should decrease with its own density; i.e., the more competitors there are the less food one gets and, through attribute 2, the fewer offspring one produces (this is usually called intraspecific competition for resources). Attribute 4: Consumers must have a finite demand for resources (appetites) and, consequently, a finite maximum reproductive rate; i.e., consumers can only eat so much before they become satiated or full. Conditions on the prey equation Condition 1: When the prey population is constant, its per-capita rate of change should decline with increasing consumer density (because of Attribute 1). Condition 2: When the predator population is constant, the per-capita rate of change of the prey should rise at first with prey density (because of Attribute 4), and then decline as prey become very dense (because of Attribute 3).

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Conditions on the predator equation (corresponding conditions for the predator R-function) Condition 3: When the prey population is constant, the per-capita rate of change of the predator should decline with increasing predator density (after Attribute 3). Condition 4: When the predator population is constant, its per-capita rate of change should rise with prey density (because of Attribute 2) to a maximum positive value (because of Attribute 4). Testing models for adherence to plausibility criteria (R function for prey) -We can see the L-V prey R-function meets condition 1 - As the L-V prey R-function does not depend on prey density (H), condition 2 is not satisfied. - We can see the predator R-function is constant when H is constant and so the L-V model violates criterion 3. - Rc rises indefinitely with prey density, condition 4 is not satisfied In other words, L-V predators are insatiable. (R function for predator)

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Logistic Model - Lotka's "fundamental equations" can be stated as follows Omly one population changes - Populations cannot grow when their densities are zero - Criterios - Grow exponentially when their densities are very small - Must decline when their densities are infinitely large a 1 = the maximum per-capita rate of change of the species in a given environment a 2 = the coefficient of intraspecific competition or, if you prefer, the "struggle for existence"

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Carrying Capacity K is often called the carrying capacity carrying capacity (w here a = a1)

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Lotka-Volterra Model with Prey Limitation Logistic Prey Growth By substituting the logistic model, a(1 - H / K) for the prey growth function (aH), we modify the prey growth rate to include intraspecific competition for resources in the lower trophic level. In other words, the prey population is limited by the "struggle for existence” in a finite environment. We can see that this model staisfies Attribute 3(considering prey as consumers), but still this logistic prey L-V model still fails 4 of the 5 conditiconditions for plausibility.

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Population Dynamics (new zero-growth isocline) (Note: if C = 0, H = K) Figure. (Left) Herbivore (H) and carnivore (C) phase space showing zero-growth isoclines for the L-V model with intraspecific competition amongst prey (prey isocline = slanting blue line, predator isocline = vertical red line). (Right) Herbivore and carnivore time series plots showing damped cycles to equilibrium.

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