Question

Consider a model for a single prey species with density N(t) given by dN/dt = r N (1 - N/K) - a N P/1 + C_1 N + C_2 P where P is a predator density and a, r, K, C_1 and C_2 are all constants. Describe in words the biological effect of the two main terms on the right hand side of the equation. Elaborate and contrast particularly the meaning of the second term in the eases when C_1 = 0, C_2 = 0, and both C_1 = 0 and C_2 = 0. Suppose C_2 = 0. Non-dimensionalize the equation and carry out an equilibrium analysis of the model, including the effect of varying levels of predator density. Discuss stability of any equilibria.

Answer 1

a)N represents prey species density that is number of species in a region and a particular time in population.

P-predator density number of predator in a population at a particular time.

K is the carrying capacity of its local environmental setting

b)dN/dt=rN(1-N/K)

equation s usaly known as Lotka–Volterra equation

Population equilibrium occurs in the model when neither of the population levels is changing, i.e. when both of the derivatives are equal to 0