III.5 Lotka-Volterra Predator-Prey Populations

III.5.1 Basic Lotka-Volterra models

Vito Volterra was an Italian mathematician who lived from 1860-1940.  His son-in-law, Humberto D'Ancona was an Italian biologist who, in 1926, completed a statistical study of fish populations in the Adriatic Sea.  D'Ancona asked Volterra if there was a mathematical model that could explain the increase in predator fish and decrease in prey fish which he observed during the World War I period.  Within a couple of months, Volterra produced a series of models for the interaction of two or more species.  Alfred J. Lotka was an American biologist and actuary who independently produced many of the same models.

The simplest of their models can be described by considering the following: What happens if our eco system contains more than one species? Additional terms are needed to describe the interactions among the different species. Obviously the interaction term must be such that no interaction takes place between two species if either of the two is extinct. The simplest expression that exhibits this property is the product of two populations! If either of the two populations is zero, the product is also zero.  In other words, the probability that the two species meet each other is proportional to the product of their population densities.

When a predator meets a prey, it gets fed and the populations exchange a certain number of calories. The predator population now has more calories and the prey population has less. Usually the Lotka-Volterra model introduces an efficiency factor, i.e. the prey population loses more calories than the predator population gains. It is also assumed that the predator population would die out when left without prey, while the prey population feeds on another species that is available in abundance and is not contained in the model.

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    the rate of decrease of prey due to encounter with predators: The foraging factor.

The simplest Lotka-Volterra system for modelling predator-prey interactions then becomes:

For a specific example, we consider the following.

In order to graph several solution curves, we make a table of solutions.

Then we can plot the solutions together.

OPTIONAL
III.5.1a Explain the behaviour of the graph in terms of temporal oscillations. Relate phase plot to the trajectory.

ADVANCED
III.5.1b During World War I many fishing boats stayed in the harbours. Fishing both on prey and on predatory species decreased enormously. Which of the four parameters a, b, c and d should be adjusted to mimic this decreased fishing pressure? Try this out. Does the Lotka-Volterra model explain D'Ancona's observations? Why?

III.5.2 Yet another Predator Prey model

The Lotka-Volterra model exhibits a number of remarkable properties. In particular, this model does not approach a continuous steady-state value. Instead, it approaches a periodic steady-state value, i.e. the solution oscillates. The shape of the oscillation is very characteristic.

There are many variations on the Lotka-Volterra equations.  Here is one example which introduces an additional quadratic term in the rate of change of y. This quadratic term reflects the meeting probability of two preys. It is often called a "self limitation" or "crowding" term. The higher the prey population, the higher their meeting probability, and the higher their competition for food and other resources will be. This competition will result in higher death rates and lower birth rates.

OPTIONAL
III.5.2a Explain the difference with the graphs you obtained in III.5.1a. Does this model explain D'Ancona's observations?