Ecology: Biology of Interaction. IV-05. The Lotka-Volterra Model
In 1925, the well-known Italian mathematician Vito Volterra, while speaking over lunch with his future son-in-law (an ichthyologist), became interested in fish population dynamics. In particular, he learned that reduced fishing during World War I increased the share of predatory fish in catches. The result was a set of models for interspecific interactions.
IV-5. The Lotka-Volterra Model In 1925, the renowned Italian mathematician Vito Volterra, while discussing with his future son-in-law, an ichthyologist by profession, became interested in fish population dynamics. Specifically, he learned that the reduction in fish catches during World War I had led to an increase in the proportion of predatory fish in the catches. The result of contemplating such facts was the models he proposed to describe interspecies interactions. "The systems studied by Volterra consist of several biological species and a food supply used by some of the species under consideration. The following assumptions are made about the components of the system. 1. Food is either unlimited or its supply over time is strictly regulated. 2. Individuals of each species die in such a way that a constant fraction of existing individuals dies per unit of time. 3. Predatory species prey on victims, and the number of victims eaten per unit of time is always proportional to the probability of encountering individuals of these two species, i.e., the product of the number of predators and the number of victims. 4. If food is unlimited and there are several species capable of consuming it, then the fraction of food consumed by each species per unit of time is proportional to the number of individuals of that species, multiplied by a coefficient that depends on the species (models of interspecies competition). 5. If a species feeds on food available in unlimited quantities, the population growth rate per unit of time is proportional to the population size. 6. If a species feeds on food available in limited quantities, its reproduction is regulated by the rate of food consumption, i.e., per unit of time, the growth rate is proportional to the amount of food consumed. The listed hypotheses allow for the description of complex living systems using systems of ordinary differential equations" (H.Yu. Ryznychenko, 1999). In essence, Volterra's models turned out to be close to the model that A. Lotka proposed in 1925 to describe the kinetics of chain chemical reactions (where the product of one reaction serves as the substrate for the next). In our textbook, we will present the Lotka-Volterra model in the form that develops the logistic model. Consider, for example, two species, A and B, which are competitors and use the same resource. We will describe the dynamics of these species using logistic equations, but we will account for both the environmental carrying capacity limitations related to resource extraction by individuals of their own species and the analogous impact from individuals of the other species. What does the multiplier (K-N)/K in the right-hand side of the logistic equation show? It shows that as the population size (N) increases, a smaller and smaller portion of the environmental carrying capacity (K) remains available for the population. But if the available resources are taken not only by individuals of one species but also by individuals of a competing species, this effect can also be accounted for in the model by introducing elements into the equation for species A that describe the impact of species B. However, species B is in an analogous situation – part of its resources are taken by individuals of species A! Since the species differ from each other, the amount of resources extracted by their individuals will be different. Let's introduce the coefficient β, which shows how many individuals of species B consume the same amount of resources as one individual of species A. Similarly, let's introduce the coefficient α, which will show how many individuals of species A consume the same amount of resources as one individual of species B. Then, using subscripts A and B to denote the values of the corresponding quantities for the two species, we can write a system of two interconnected equations. The Lotka-Volterra model played an exceptional role in the development of mathematical ecology. As can be easily understood, based on it, numerous other, more complex models can be built. For example, they can describe the interaction of not two, but a larger number of resources. The parameter K for each species may be constant, or it may change according to some law (e.g., depending on weather changes or seasonal changes). The reaction of one species to changes in the population size of another may occur with a greater or lesser delay, and so on. The simple equations presented here are a rather powerful tool for studying natural processes!