Question

How modeling trophic interactions using the Lotka- Volterra equations can produce population cycles in predator-prey dynamics ( using arguments based on isoclines) ?

Ecology subject

Answer 1

Modelling trophic interactions with features of the environment by considering a very simple model of a predator feeding on a single prey species. This model was first proposed independently by Alfred Lotka in 1925 and Vito Volterra in 1926.

By stating the using arguments and then turning into mathematics are:-

Two species are being modeled a predator will be denoted as P and the abundance of the prey will be denoted as N. The abundances of the two species will respond to one another instantaneously and so their abundances will be modeled in continuous time. Differential equations to represent the changes in their abundances with time : dP\dT and dN\dT. In the absence of predators the prey species follows exponential population growth is dN\dT=rN. The most simple model of encounters here is called mass action which includes attack coefficient, type I functional response, the full prey equation is dN\dT=rN-aNP, the rate at which the predator population produces babies are simply the product of how many prey are aNP times the number of babies produced for each prey killed which ic baNP. Now the full equation is dP\dT=baNP-dP

The coupling is based on how the various demographic rates depend on the abundances of the two species like N's and P's in each equation.This is what couples their population dynamics. These are called zero net growth isoclines. The isoclines function will be very important in understanding the dynamics and stability of the model. The prey isoclines is given by the functions, P=r\a and the predator isoclines is given by the function N=d\(ab) . Each function maps out the abundance of that species when it will not change in abundance.A good model must be simple enough to be mathematically tractable, but complex enough to represent a system realistically.Realism is often sacrificed for simplicity , and one of the shortcomings of the Lotka-Volterra models its reliance on unrealistic assumptions.For example, prey population are limited by food resources and not just by predation, and no predator can consume infinite quantities of prey which describes cyclical relationship .