Question

Consider the following model of a food chain where species X is a predator of species Y which is a predator of species Z. Let x(t),y(t), and z(t) represent the densities of the three populations respectively at time t. A density value of zero would mean that the corresponding population has died out. The equations describing the growth of each population density is (the values of the densities are non-negative initially): x′=−rx+axy y′=−sy−bxy+cyz where the z′=qz−dyz parameters a,b,c,d,r,s,q are all positive values .Is this system linear? Explain. If the prey (Z) were to die out what would happen to the Y and Z species in the long run?

Answer 1

The system describing the predator-prey interactions is not linear because of the product terms involving species densities. If the prey Z dies out, species Y and X would likely follow due to the dependence of their growth on the prey's density.

Step-by-step explanation:

The given system of equations is not linear because it includes terms that involve the product of two variables (e.g., axy, bxy, and cyz). Linear systems have the property that their rate of change is constant or depends linearly on the variables.

The presence of product terms means that the rate of change of one species depends on the density of both itself and another species, which is a hallmark of nonlinear systems.

If the prey (Z) were to die out (z(t)=0), meaning a density of zero, this would have a cascading effect on the food chain: Species Y, which relies on Z for food, would have no food source and would likely die out in the long run due to the term cyz in its growth rate equation becoming zero. Subsequently, as species Y's density decreases, species X would also suffer due to lack of food (reflected by the bxy term in its growth rate equation), potentially leading to its die-out as well.