A model is proposed to understand the dynamics in a food chain (one predator‐two prey). Unlike many approaches, we consider mutualism (for defense against predators) between the two groups of prey. We investigate the conditions for coexistence and exclusion. Unlike Elettreby's (2009) results, we show that prey can coexist in the absence of predators (as expected since there is no competition between prey). We also show the existence of Hopf bifurcation and limit cycle in the model, and numerically present bifurcation diagrams in terms of mutualism and harvesting. When the harvest is practiced for profit making, we provide the threshold effort value that determines the profitability of the harvest. We show that there is zero profit when the constant effort is applied. Below (resp. above) , there will always be gain (resp. loss). In the case of gain, we provide the optimal effort and optimal steady states that produce maximum profit and ensure coexistence.

中文翻译：

---

具有相互防御的食物链模型中的共存和收获控制策略

提出了一个模型来理解食物链中的动态（一个捕食者-两个猎物）。与许多方法不同，我们考虑了两组猎物之间的互惠关系（以防御天敌）。我们调查了共存和排斥的条件。与Elettreby（2009）的结果不同，我们表明猎物可以在没有捕食者的情况下共存（如预期的那样，因为猎物之间没有竞争）。我们还显示了模型中Hopf分叉和极限环的存在，并在互惠和收获方面以数值形式显示了分叉图。当实践收获以牟利时，我们提供确定收获利润的阈值努力值。我们证明了不断努力的结果是零利润。低于（高于上面），总会有收益（亏损）。在收益的情况下，我们提供了最佳的努力和最佳的稳定状态，从而产生最大的利润并确保共存。