Consider the following threespecies ecosystem dFdt Fa cS d

Consider the following three-species ecosystem: dF/dt = F(a - cS) dS/dt = S(-k + lambda F = mG) dG/dt = G (-e + sigma S). Assume that the coefficients are positive constants. Describe the role each species plays in this ecological system.

Solution

1st one ::

solving by variabe seperable

dF/F(a-cS) = dt

=> integrating both sides

=> 1/(a-cS)lnF = t +c1

=> lnF = (a-cS)(t +c1)

=>F = e^((a-cS)(t +c1))

therefore since function is exponential therefore number of species increases with time

2nd one

dS/S(-k +lamdaF-mG) = dt

=> same as above

=> 1/S(-k +lamdaF-mG) ln S = t +c2

=> S = e ^((S(-k +lamdaF-mG) )(t+c2))

therefore since function is exponential therefore number of species increases with time

3rd one

dG/G(-e+sigma S) = dt

=> same as above

=> G = e^(((-e+sigma S))(t+c3))

therefore since function is exponential therefore number of species increases with time