Question: Variation of Predator-Prey Model
Sunday, 11 of November , 2007 at 6:25 pm
The Lotka-Volterra Predator-Prey Model attempts to describe the dynamics of a biological system where two species interact, one a predator and one its prey. The model is described by the following first order, non-linear, differential equations:
where
- x = number of prey
- y = number of predators
- a = growth rate of the prey
- c = death rate of the predator
= predation rate coefficient
= reproduction rate of predators per 1 prey eaten
Question: A major flaw in this model is that in the absence of a predator, the prey would grow without bound. If we assume that the population growth of the prey reduces to a logistic equation in the absence of a predator, we can get the following variation of Lotka-Volterra’s differential equation.
where 
is the rate of saturation of the prey.
Find the critical points and discuss their nature and stability characteristics.
The Lotka-Volterra Predator-Prey Model attempts to describe the dynamics of a biological system where two species interact, one a predator and one its prey. The model is described by the following first order, non-linear, differential equations:
where
- x = number of prey
- y = number of predators
- a = growth rate of the prey
- c = death rate of the predator
- = predation rate coefficient
- = reproduction rate of predators per 1 prey eaten
Question: A major flaw in this model is that in the absence of a predator, the prey would grow without bound. If we assume that the population growth of the prey reduces to a logistic equation in the absence of a predator, we can get the following variation of Lotka-Volterra’s differential equation.
where
is the rate of saturation of the prey.
Find the critical points and discuss their nature and stability characteristics.
Category: Differential Equations, Questions
Comment by Lane
Made Tuesday, 13 of November , 2007 at 4:54 pm
You meant for the RH side of the first equation to be contain sigma*x not sigma*y, right?
Comment by eldila
Made Tuesday, 13 of November , 2007 at 7:29 pm
yep, good catch. I will change that.
Comment by Lane
Made Wednesday, 14 of November , 2007 at 1:31 pm
This is a great problem. I think it is beyond my skills, but I am taking a whack at it. I look forward to seeing to seeing the solution.
Comment by eldila
Made Wednesday, 14 of November , 2007 at 2:18 pm
I am looking for characteristics of the differential equation, so don’t try to solve it exactly.
ie. There is a critical point at (0,0) which is a saddle point. You can only get to it if x=0.