where z is a positive parameter that models predator interference (Figure 3). Several other types of functional responses can be found in the literature.
Gause, A. N. Kolmogorov, W. W. Murdoch, and others to analyze prey-predator models where the linear functional and numerical responses of the Lotka-Volterra model are replaced by more general functions. A general representation of a prey-predator model is dR
where r(R) is the per capita prey growth rate, f (R,C) is the functional response, g (R,C) is the numerical response, and m is the per capita predator mortality rate. For r(R) = r, f (R,C) = AR and g(R,C) = eAR, the above model coincides with the Lotka-Volterra model .
In what follows we will assume that model  has a single positive equilibrium R* and C. Then the question is, what is the long-term behavior of prey and predator abundances? Do they converge to this equilibrium? The usual starting point to answer this question is to study conditions under which the equilibrium is locally asymptotically stable. Conditions that guarantee local asymptotic stability of the equilibrium are given in terms of the Jacobian matrix evaluated at the equilibrium of model :
To derive the above matrix we used the fact that at the equilibrium, g(R*,C*) = m. If the sum of the two diagonal elements (i.e., the trace) of the Jacobian matrix is negative and the determinant is positive then the equilibrium is locally asymptotically stable. This leads to the following two general conditions:
Although these two conditions look quite formidable, they will substantially simplify for particular cases of functional and numerical responses considered in the next section.
Was this article helpful?
Do You Want To Learn More About Green Living That Can Save You Money? Discover How To Create A Worm Farm From Scratch! Recycling has caught on with a more people as the years go by. Well, now theres another way to recycle that may seem unconventional at first, but it can save you money down the road.