Modeling Competitive Interactions Between Two Species Using Logistic Growth

Problem

Two populations coexist in a natural environment with relationships: competition, mutual dependence, or predator‑prey.

When they compete for the same food and space, the typical outcome is extinction of the weaker competitor and the stronger reaching the environment’s carrying capacity.

Establish a mathematical model describing the competitive process and analyze the conditions leading to this outcome.

Model Assumptions

Populations A and B each follow logistic growth when isolated.

When coexisting, the inhibitory effect of B on A’s growth is proportional to B’s size, and vice versa.

Model

For the resources consumed by population A, population B exerts a larger inhibitory effect, indicating that B has stronger competitive ability.

Various Forms of Two‑Population Models

Competition

Mutual dependence

Predator‑prey (the strong eat the weak)