Prey-predator model

[matematician_best]

Gause-Rosenzweig-McArthur models for predator-prey interactions, featuring a logistic growth rate for the prey, possess the subsequent configuration:
H'(t) = r(1-H/K)H-f(H)P
P'(t) = -uP+gf(H)P
where r, u and g are constants, f(H) is the functional response (number of preys that a predator eats in a unit of time).
Suppose f(H) increasing, such that f(0)=0 and that the limit of f(H), as H approaches infinity, is a finite number, that we indicate with m. Show graphically that, if gm>u, then there exists a unique value H* such that gf(H*)=u. Compute the isoclines of the system. Show that the system has a positive equilibrium if and only if K>H*.

 

[ROBERTMILLS]

The Gause-Rosenzweig-McArthur models for predator-prey interactions, with a logistic growth rate for the prey, are governed by the equations:

H′(t)=r(1−H/K)H−f(H)P

P′(t)=−uP+gf(H)P

where r, u, and g are constants, and f(H) represents the functional response, indicating the number of preys a predator consumes in a unit of time.

Considering f(H) to be an increasing function, with f(0)=0 and limH→∞f(H)=m, where m is a finite number. We aim to demonstrate graphically that if gm>u, then there exists a unique value H∗ such that gf(H∗)=u.

This scenario implies that the rate at which predators consume prey gf(H)) surpasses the rate at which predators die (u). Graphically, this condition leads to an intersection between the prey growth isocline (r(1−H/K)H) and the prey consumption isocline (gf(H)P), signifying an equilibrium point.

The equilibrium point H∗ corresponds to the intersection of these two isoclines, where the rate of prey consumption by predators matches the prey's growth rate. By solving gf(H∗)=u, we can find this unique value H∗.

Furthermore, to illustrate the behavior of the system, we can plot the isoclines of the system. These isoclines represent the trajectories along which the populations of prey and predators remain constant over time. They reveal critical points such as stable equilibria, where both populations stabilize, and unstable equilibria, where slight perturbations lead to population fluctuations.

The system exhibits a positive equilibrium when K>H∗, indicating that the carrying capacity of the environment for prey exceeds the equilibrium population size. This condition ensures sustainable growth and stability for both predator and prey populations.

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