Spatiotemporal Dynamics of Competing Species With or Without Memory Under Dirichlet Boundary Condition

ABSTRACT We investigate a diffusive Lotka–Volterra competition model with temporally distributed memory and Dirichlet boundary conditions, focusing on the interaction between a species with memory and one without. The memory‐capable species exhibits both self‐memory and cross‐memory, while the memoryless species relies solely on random diffusion. We analyze the existence and stability of steady‐state solutions, including semi‐trivial and positive steady states, under two distinct memory kernel cases. In the weak kernel case, where memory fades over time after immediate acquisition, the positive steady‐state solution remains locally asymptotically stable for all non‐negative delays. In the strong kernel case, where memory follows both an acquisition and decay phase, Hopf bifurcations arise as delay increases, leading to instability and the emergence of nonhomogeneous periodic solutions. Our findings reveal that species with self‐memory gain a competitive advantage, increasing their likelihood of survival, while those relying solely on cross‐memory face a higher risk of extinction. This contrast underscores the crucial role of different memory types in shaping competitive outcomes.