Determine the equilibrium points of a predator-prey model?

Question

I am trying to determine the equilibrium points

r=1;
b=1;
c=0.01;
μ=0.4;
deq1 = x'[t] == r*x[t] - b*x[t]^2 - c*x[t]*y[t] - 0.75*x[t]*y[t]/(a + x[t]) 
deq2 = y'[t] == μ*y[t] + 0.75*x[t]*y[t]/(a + x[t]) 

I used the following code to determine the equilibrium points :

equilibrio = NSolve[{r*x[t] - b*x[t]^2 - c*x[t]*y[t] - 0.75*x[t]*y[t]/(a + x[t])  == 0, -μ*y[t] + 0.75*x[t]*y[t]/(a + x[t])  == 0}, {x[t], y[t]}]

And i get the following error: Solve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result.

The question is, have I made any mistakes, or does the exercise have to define the parameter 'a'?

Answer 1

Rationalize your equation and (N)Solve works as expected without message:

equilibrio = Solve[{r*x[t] - b*x[t]^2 - c*x[t]*y[t] - 0.75*x[t]*y[t]/(a + x[t])== 0, -μ*y[t] + 0.75*x[t]*y[t]/(a + x[t])  == 0}//Rationalize[#,0]&, {x[t], y[t]}]
(*{{x[t] -> 1., y[t] -> 0}, {x[t] -> 1.14286 a, 
y[t] -> -((14.2857 (-7. a + 8. a^2))/(35. + a))}, {x[t] -> 0., 
y[t] -> 0.}}{{x[t] -> 1, y[t] -> 0}, {x[t] -> (8 a)/7, 
y[t] -> -((100 (-7 a + 8 a^2))/(7 (35 + a)))}, {x[t] -> 0,y[t] -> 0}}*)