Ecology: The Biology of Interactions. 4.05. The Lotka-Volterra Model
The Lotka-Volterra model played an exceptional role in the development of mathematical ecology. As is easy to see, many other, more complex models can be built on its basis. For example, they can describe relationships not between two but among a larger number of resources. The parameter K for each species may remain...
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4.04. Exponential and Logistic Population Growth
D. Shabanov, M. Kravchenko. Ecology: Biology of Interaction. Chapter 4. Population Ecology
4.06. Classification of Population Interactions
4.05. The Lotka-Volterra Model In 1925, the well-known Italian mathematician Vito Volterra, while talking over lunch with his future son-in-law, an ichthyologist by profession, became interested in fish population dynamics. For example, he learned that reduced fish catches during World War I led to an increase in the proportion of predatory fish in catches. The result of reflecting on such facts was the set of models he proposed for describing interspecific interactions. "The systems studied by Volterra consist of several biological species and a food reserve used by some of the species considered. The following assumptions are made about system components. 1. Food is either available in unlimited quantities, or its inflow over time is rigidly regulated. 2. Individuals of each species die in such a way that, per unit time, a constant fraction of the existing individuals dies. 3. Predatory species consume prey, and per unit time the number of prey consumed is always proportional to the probability of encounter between individuals of the two species, i.e., to the product of predator abundance and prey abundance. 4. If there is unlimited food and several species able to consume it, then the share of food consumed by each species per unit time is proportional to the abundance of that species multiplied by a species-specific coefficient (models of interspecific competition). 5. If a species feeds on food available in unlimited quantity, the increase in abundance per unit time is proportional to the abundance of that species. 6. If a species feeds on food available in limited quantity, then its reproduction is regulated by the rate of food consumption, i.e., per unit time the increase is proportional to the amount of food consumed. These hypotheses make it possible to describe complex living systems using systems of ordinary differential equations" (G. Yu. Riznichenko, 1999). In essence, Volterra's models were close to the model proposed by Lotka in 1925 for describing kinetics of chain chemical reactions (where the product of one reaction serves as substrate for the next). In this textbook we present the Lotka-Volterra model in the form that extends the logistic model. Consider, for example, two species, A and B, that are competitors and use the same resource. Let us describe dynamics of these species by logistic equations, while accounting both for carrying-capacity limitations associated with resource withdrawal by individuals of their own species and for analogous effects from individuals of the competing species. What does the factor in the right-hand side of the logistic equation, (K-N)/K, indicate? It shows that as abundance (N) grows, a smaller and smaller fraction of carrying capacity (K) remains available to the population. But if available resources are withdrawn not only by individuals of one species, but also by individuals of a competing species, this effect can also be incorporated by introducing into the equation for species A terms describing the influence of species B. Species B is in an analogous position: part of its resources is withdrawn by individuals of species A. Since species differ, the amount of resources withdrawn by their individuals will differ. Let us introduce coefficient β showing how many individuals of species B consume the same quantity of resources as one individual of species A. Similarly, coefficient α will show how many individuals of species A consume the same quantity of resources as one individual of species B. Then, denoting corresponding values for the two species with subscripts A and B, we can write a system of two coupled equations. [IMG_1] The Lotka-Volterra model played an exceptional role in the development of mathematical ecology. As is easy to see, many other, more complex models can be built on its basis. For example, they may describe interrelations not between two, but among a larger number of resources. Parameter K for each species may be constant, or may vary according to some law (for example, depending on weather change or seasonal shifts). The response of one species to changes in abundance of the other may occur with greater or lesser delay, etc. The relatively simple equations presented here are a sufficiently powerful tool for studying natural processes. Additional material: Educational model: Lotka-Volterra predator-prey model
4.04. Exponential and Logistic Population Growth
D. Shabanov, M. Kravchenko. Ecology: Biology of Interaction. Chapter 4. Population Ecology
4.06. Classification of Population Interactions