Predator self-competition and prey mobility jointly set spatial structure in an additional-food predator-prey system: weak competition gives whole-field oscillations, stronger competition gives fixed patches, and near the crossover the two combine into pulsing patterns

In the well-mixed setting the authors locate the Hopf bifurcation of the coexistence state exactly and show the cycle born there is stable, so weak competition corresponds to boom-bust oscillations rather than unbounded growth. Prior discussion of additional-food biological control noted that with nothing limiting the predator's own numbers the extra food lets its population grow without bound; this work brings predator intraspecific competition into the model as that missing brake and gives an analytical conclusion about the stability of the oscillation. Based on a reaction-diffusion model of a logistically growing prey and a predator feeding through a Holling type II response that also draws on additional food, with the Hopf bifurcation of the coexistence state solved exactly and the stability of the resulting cycle determined in the well-mixed setting.

Allowing movement, the authors obtain the diffusion-driven Turing threshold at which the uniform state breaks into stationary patches of high and low density, and find the uniform oscillation stable as it appears. How predator self-limitation reshapes the spatial arrangement of the two species had not been asked; this work brings Turing analysis into a system with additional food and predator intraspecific competition and gives the threshold condition for patch formation. Derived through linear stability analysis and the Turing threshold of the reaction-diffusion model, together with numerical simulations confirming that the uniform state breaks into stationary patches.

With prey mobility and competition strength as control parameters, the Turing-type (pattern-forming) and oscillatory instabilities meet at a single crossover point, where the authors compute the dynamics. Placing the two instability types on one parameter plane and locating their intersection makes competition strength and prey spread joint controls that distinguish spatial patterns from temporal oscillations. The boundaries of the two instabilities are solved in the parameter space spanned by prey mobility and competition strength, their intersection is located, and the dynamics are then computed at that point.

Simulations confirm the sequence: weak competition gives a whole-field oscillation, stronger competition with faster prey spread gives fixed patterns, and near the crossover the two combine into patterns that pulse in time. The analytically obtained thresholds and crossover point are translated into an observable sequence of three spatio-temporal behaviors, showing that predator self-competition sets the spatial structure of the community. Numerical simulations validate the analytical predictions across three cases: weak competition, stronger competition with faster prey spread, and the neighborhood of the crossover.

The results are aimed at augmentative biological control settings where additional food is used to steer a released predator, and apply to reaction-diffusion systems composed of a logistically growing prey and a predator with a Holling type II response that also draws on additional food, with intraspecific competition among predators. They let researchers treat competition strength and prey spread as two adjustable knobs to anticipate whether the system moves toward whole-field oscillation, fixed patches, or spatio-temporal pulsing patterns, thereby guiding additional-food release strategies in theory.

The text is a full read at the abstract level and provides no specific parameter values, simulation grids, or figures, so the exact location of the crossover and the quantitative boundaries of the three regimes cannot be confirmed from the available wording. In addition, the model assumes additional food does not reproduce and represents predator competition with a single strength parameter; how well these assumptions hold in real field systems, and how robust the conclusions are to more complex functional responses or spatial heterogeneity, remain open questions worth following up.