Stability Analysis of a Discrete Dynamical System With Density Dependence and Mutual Interference

Abstract

To address the limitations of classical fishery population models (e.g., Ricker and Beverton–Holt) in capturing complex interspecific interactions and lag effects, this article constructs an improved two‐species, discrete dynamic model, y n + 1 = a y n 1 + p z n e − y n , z n + 1 = b z n 1 + q y n e − z n $$ {y}_{n+1}=\frac{a{y}_n}{1+p{z}_n}{e}^{-{y}_n},\kern1em {z}_{n+1}=\frac{b{z}_n}{1+q{y}_n}{e}^{-{z}_n} $$ where the parameters a , b , p , q $$ a,b,p,q $$ are positive real numbers and the initial values y 0 , z 0 $$ {y}_0,{z}_0 $$ are arbitrary nonnegative real numbers. We rigorously analyze the dynamic behaviors of the system, establishing that positive solutions are uniformly bounded and strictly persistent under specific intrinsic growth conditions ( a > 1 $$ a>1 $$ and b > 1 $$ b>1 $$ ). Furthermore, we provide a comprehensive classification of the boundary dynamics, exploring non‐hyperbolic and 1:1 resonant scenarios. We derive sufficient conditions for global asymptotic stability (GAS): (1) the trivial fixed point (extinction) is GAS via direct inequality bounding when intrinsic growth rates satisfy a < 1 $$ a<1 $$ and b < 1 $$ b<1 $$ , and (2) utilizing the mean value theorem (MVT) and Lyapunov function methods, the unique positive fixed point (coexistence) is GAS when growth rates lie within the non‐chaotic range ( 1 < a , b ≤ e $$ 1