# Lotka-Volterra model, nonlinear Ordinary Differential equations

#### sapiental

1. The problem statement, all variables and given/known data

The following set of nonlinear Ordinary Differential equations is known as the Lotka-Volterra modek:

dx/dt = (a - bx - cy)x
dy/dt = (-d+ex)y

where a,b,c,d,e are positive constants. These equations model a simple eco system of preator and pray. the x is hares and y is foxes in a forest. Describe the physical meaning of the number of hares and foxes in a forest.

a = 10, b = 10^-5, c =.1, d = 10, e =.1

2. Relevant equations

dx/dt = (a - bx - cy)x
dy/dt = (-d+ex)y

3. The attempt at a solution

I think...

a = the coefficient for the increase in the mice population due to resources
b = the natural death coefficient rate of mice not due to predators
c = the coefficient for the the amount of mice get eatean based on population size
d = the coefficient for the natural deaths based on predator population
e = coefficient for the effect of amount of predators vs prey

#### vanesch

Staff Emeritus
Gold Member
1. The problem statement, all variables and given/known data

The following set of nonlinear Ordinary Differential equations is known as the Lotka-Volterra modek:

dx/dt = (a - bx - cy)x
dy/dt = (-d+ex)y

where a,b,c,d,e are positive constants. These equations model a simple eco system of preator and pray. the x is hares and y is foxes in a forest. Describe the physical meaning of the number of hares and foxes in a forest.

a = 10, b = 10^-5, c =.1, d = 10, e =.1

2. Relevant equations

dx/dt = (a - bx - cy)x
dy/dt = (-d+ex)y

3. The attempt at a solution

I think...

a = the coefficient for the increase in the mice population due to resources
b = the natural death coefficient rate of mice not due to predators
c = the coefficient for the the amount of mice get eatean based on population size
d = the coefficient for the natural deaths based on predator population
e = coefficient for the effect of amount of predators vs prey

Think again. The coefficient a gives you how much hares (not mice!) become into being, proportional to the number of existing hares. What could that be ?

The coefficient b gives you the decrease of the number of hares, due to the SQUARE of the number of existing hares. If it were the death rate, it would simply be proportional. So it is the INCREASE in death rate, due to the presence of other hares. What could that be ? In what way does the presence of other hares increase the death rate ?

The coefficient c gives you the death rate as proportional to the number of foxes. What could that be ?

Your explanation of d seems ok.

Your explanation of e should be more explicit. Of course it is a coefficient. What does it mean ?

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