3 species Lotka–Volterra model. Limit cycle

Question

Good day, I have 3 species Lotka–Volterra model. My goal is to determine if there is a limit cycle in the system

$$ \left\{ \begin{array}{l} \frac{d c}{d t}=r_c c(1-c)-\frac{c h}{c+\theta_1} \\ \frac{d n}{d t}=r_n n(1-n)-\frac{n h}{n+\theta_2} - \varepsilon c n \\ \frac{d h}{d t}=-\gamma h+h\left(k_1 c+k_2 n\right) \end{array} \right. $$

I have found 6 equilibrium points:

$$ A_1=(0; 0; 0), A_2=(1; 1-\frac{\varepsilon}{r_n}; 0), A_3=(0; 1; 0), A_4=(1; 0; 0), A_5=(0; \frac{\gamma}{k_2} ; r_n\left(1-\frac{\gamma}{k_2}\right)\left(\frac{\gamma}{k_2}+\theta_2\right)), A_6=(\frac{\gamma}{k_1}; 0;r_c\left(1-\frac{\gamma}{k_1}\right)\left(\frac{\gamma}{k_1}+\theta_1\right) \) $$

Last point is unkown: $$ A_7=(c^{\star}; n^{\star}; h^{\star}); $$

I try to find it this way: $$ h=r_c(1-c)\left(c+\theta_1\right) $$ $$ h=\left(r_n(1-n)-\varepsilon c\right)\left(n+\theta_2\right) $$ $$ n=\frac{\gamma-k_1c}{k_2} $$ $$ r_c(1-c)\left(c+\theta_1\right) = \left(r_n(1-\frac{\gamma-k_1c}{k_2})-\varepsilon c\right)\left(\frac{\gamma-k_1c}{k_2}+\theta_2\right) $$

but i have very big expression for c and n and it's hard to determine type of equilibrium point putting it into jacobian. I built phase portrait and it shows that there limit cycle in point (c,n,h). I don't know am I doing it right way, and how to prove that there is limit cycle.

Answer 1

You can find the characteristic polynomial of the Jacobian evaluated at the coexistence equilibrium, and look at the Hurwitz determinants.

• If $H_1, H_2, H_3 > 0$ then you will have local asymptotic stability
• For parameter of interest $\alpha$ at point $\alpha_0$ if $H_1(\alpha_0)$ > 0, $H_2(\alpha_0) = 0$ ($\Rightarrow H_3(\alpha_0) = 0$) and $\frac{d H_2}{d\alpha}(\alpha_0) \neq 0$ then there is a simple Hopf bifurcation in parameter $\alpha$ at point $\alpha_0$

This second point implies that you have a pair of purely imaginary eigenvalues at point $\alpha_0$. This will result in either stable or unstable limit cycles

In general the asymptotic behavior of solutions near this equilibrium will depend on the choice of parameters. This is where bifurcation theory comes in useful. Given your choices of functional form I would expect that this model displays either Hopf bifurcation or trans-critical bifurcation dependent on parameter value choices.

Also worthwhile is to provide conditions under which the coexistence equilibrium exists