"An example of a differential equation that serves as a slightly more realistic model

of population growth than the exponential growth model is the Lotka-Volterra DE,

dy/dt= ky − ay^2, (1)

Once again, y(t) denotes the population of a species at time t. The term −ay^2, which decreases the rate of growth, represents

competition between population members for resources. In the special case a = 0, there is no competition, resulting in exponential growth.

(a) Show that the constant population y(t) = k/a is a solution to Eq. (1).

(b) Show that the following function, y(t) = y0/[(1 − by0)e^(−kt) + by0], t >= 0, (2)

where b = a/k, is a solution to DE in (1). Also show that it satisfies the

(positive) initial condition y(0) = y0 > 0"

I've been working on this question for what seems like an eternity and I just can't fully understand how to prove. I need a hint, or something, please someone give me a push in the right direction.

of population growth than the exponential growth model is the Lotka-Volterra DE,

dy/dt= ky − ay^2, (1)

Once again, y(t) denotes the population of a species at time t. The term −ay^2, which decreases the rate of growth, represents

competition between population members for resources. In the special case a = 0, there is no competition, resulting in exponential growth.

(a) Show that the constant population y(t) = k/a is a solution to Eq. (1).

(b) Show that the following function, y(t) = y0/[(1 − by0)e^(−kt) + by0], t >= 0, (2)

where b = a/k, is a solution to DE in (1). Also show that it satisfies the

(positive) initial condition y(0) = y0 > 0"

I've been working on this question for what seems like an eternity and I just can't fully understand how to prove. I need a hint, or something, please someone give me a push in the right direction.

Last edited: