The steady state bifurcation and spatiotemporal patterns are induced by prey-taxis in a population model, in which prey, predators and scavengers are involved. Effects of prey-taxis are manifested from the obtained results. By using the prey-taxis coefficient as the bifurcation parameter, the occurrence conditions of the steady state bifurcation is established. It is found that there is no steady state bifurcation without prey-taxis or with the attractive type prey-taxis, meanwhile, the steady state bifurcation will occur when the repulsive type prey-taxis is present. In the sequel, by employing the Crandall–Rabinowitz local bifurcation theory, the existence of the bifurcating solution and its stability are further explored. It is stable if the second-order perturbation term is less than zero, and unstable if greater than zero. The resulting nonconstant steady states are displayed through numerical simulation, which are in agreement with theoretical analysis. No steady state bifurcation or patterns will appear in simulation when the prey-taxis is absent.

中文翻译：

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扩散群体模型的稳态分岔和模式形成

稳态分叉和时空模式是由种群模型中的猎物趋向性引起的，其中涉及猎物、捕食者和食腐动物。猎物趋向性的影响从所获得的结果中可见一斑。以prey-taxis系数作为分岔参数，建立了稳态分岔的发生条件。研究发现，在没有猎物趋向性的情况下，或者有吸引型猎物趋向性的情况下，都不存在稳态分岔，而当存在排斥型猎物趋向性时，就会出现稳态分岔。接下来，利用Crandall-Rabinowitz局部分岔理论，进一步探讨了分岔解的存在性及其稳定性。如果二阶扰动项小于零，则稳定；如果大于零，则不稳定。通过数值模拟显示了所得的非恒定稳态，这与理论分析是一致的。当猎物出租车不存在时，模拟中不会出现稳态分叉或模式。