This week we worked on understanding in detail the behavior of one of the steady state solutions for Model 2s (the model where both producers and scroungers die linearly and the scroungers depend on the producers to eat):

P → ( r ( ( k / β ) – δ_1) ) / ( b k / β ) S → 0 F → δ_1 / β

We found that:

1) There will never be a case in which the radical in the eigenvalue is a real number and the entire eigenvalue is negative.

2) For almost all cases the radical will be imaginary and therefore the entire eigenvalue will be negative

3) The only way that the radical would be real is if δ_1 is greater than (4 k^2 / β^2) / (r+4 k /β)

4) For the third eigenvalue to be negative, δ_1 >= k /β which is very unlikely to occur because β would have to be very small

5) It could be possible that the first two eigenvalues are positive if δ_1 >= k \ β

6) Our questions

– How are all three eigenvalues sometimes negative?