12.6: Independent carrying capacities

Self-limitation or self-enhancement of population growth are within the $$r\,+\,sN$$ framework. Below these terms are in red.

$$\frac{1}{N_1}\frac{dN_1}{dt}\,=\,r_1\,s_{1,2}N_2\,\color{red}{+\,s_{1,1}N_1}$$

$$\frac{1}{N_2}\frac{dN_2}{dt}\,=\,r_2\,s_{2,1}N_1\,\color{red}{+\,s_{2,2}N_2}$$

The self-feedback term for the prey, $$s_{1,1}$$, is typically negative, reflecting a carrying capacity for the prey in the absence of predators, $$K_1\,=\,-r_1\,/\,s_{1,1}$$. This tends to stabilize the system, dampening oscillations and leading to a joint equilibrium of predator and prey.

On the other hand, the self-feedback term for the predator, $$s_{2,2}$$, is typically zero, meaning the predators vanish in the absence of prey. But it could be positive, indicating benefits from group hunting and the like. A positive value for $$s_{2,2}$$ tends to destabilize the system, leading to enlarging oscillations.