In this paper, we discretize and analyze a stoichiometric optimal foraging model where the grazer's feeding effort depends on the producer's nutrient quality. We systematically make comparisons of the dynamical behaviors between the discrete-time model and the continuous-time model to study the robustness of model predictions to time discretization. When the maximum growth rate of producer is low, both model types admit similar dynamics including bistability and deterministic extinction of the grazer caused by low nutrient quality of the producer. Especially, the grazer is benefited from optimal foraging similarly in both discrete-time and continuous-time models. When the maximum growth rate of producer is high, dynamics of the discrete-time model are more complex including chaos. A phenomenal observation is that under extremely high light intensities, the grazer in the continuous-time model tends to perish due to poor food quality, however, the grazer in the discrete-time model persists in regular or irregular oscillatory ways. This significant difference indicates the necessity of studying discrete-time models which naturally include species' generations and are thus more popular in theoretical biology. Finally, we discuss how the shape of the quality-based feeding function regulates the beneficial or restraint effect of optimal foraging on the grazer population.

中文翻译：

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离散时间化学计量最优觅食模型的动力学

在本文中，我们离散化并分析了化学计量的最佳觅食模型，其中放牧者的进食量取决于生产者的营养素质量。我们系统地比较了离散时间模型和连续时间模型之间的动力学行为，以研究模型预测对时间离散的鲁棒性。当生产者的最大生长率较低时，两种模型类型都具有相似的动力学特性，包括双稳态和由生产者营养素质量低引起的放牧者的确定性灭绝。尤其是，无论是在离散时间模型还是连续时间模型中，均可以从最佳觅食中受益。当生产者的最大增长率很高时，离散时间模型的动力学会更加复杂，包括混乱。令人惊讶的观察是，在极高的光照强度下，连续时间模型中的放牧者由于食物质量差而趋于灭亡，但是，离散时间模型中的放牧者以规则或不规则的振荡方式持续存在。这一显着差异表明研究离散时间模型的必要性，该模型自然包括物种的世代，因此在理论生物学中更受欢迎。最后，我们讨论了基于质量的进食功能的形状如何调节最佳觅食对草食者种群的有利或制约作用。这一显着差异表明研究离散时间模型的必要性，该模型自然包括物种的世代，因此在理论生物学中更受欢迎。最后，我们讨论了基于质量的进食功能的形状如何调节最佳觅食对草食者种群的有利或制约作用。这一显着差异表明研究离散时间模型的必要性，该模型自然包括物种的世代，因此在理论生物学中更受欢迎。最后，我们讨论了基于质量的饲喂功能的形状如何调节最佳觅食对草食者种群的有利或抑制作用。