Dynamical Analysis of a Discrete Amensalism System with the Beddington-DeAngelis Functional Response and Allee Effect for the Unaffected Species
具有贝丁顿-德安吉利斯功能响应和未受影响物种的阿利效应的离散同化系统的动力学分析

Received: 27 September 2022 / Accepted: 2 December 2022 / Published online: 15 December 2022 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022
收到：2022 年 9 月 27 日 / 接受：2022 年 12 月 2 日 / 在线发表：2022 年 12 月 15 日 © 作者，独家授权施普林格自然瑞士股份公司 2022 年出版

Keywords Amensalism model • Beddington-DeAngelis functional response • Allee effect • Bifurcation
关键词 Amensalism 模型 - Beddington-DeAngelis 功能响应 - 阿利效应 - 分岔

Species are constantly interacting with one another in nature. Many biological processes in ecosystems, such as the food chain and the nutrient cycle, are based on species interactions. Competition and predation, commensalism, parasitism, mutualism, and amensalism are the five relationships between species. In particular, amensalism is a sort of biological relationship in which one species delivers harm to another without incurring any costs or advantages. Though it may not be extensively investigated as other interaction forms, amensalism is not rare. For example, Spanish ibex shows
在自然界中，物种之间不断发生相互作用。生态系统中的许多生物过程，如食物链和营养循环，都是基于物种之间的相互作用。竞争与捕食、共生、寄生、互生和补偿是物种之间的五种关系。其中，补偿关系是一种生物关系，在这种关系中，一个物种会给另一个物种带来伤害，而不会产生任何代价或好处。虽然它可能不像其他互动形式那样被广泛研究，但 "补偿主义 "并不罕见。例如，西班牙山羊表现出

amensal relationship with weevils, which feeds on the same type of plants as ibex. The presence of ibex has a terrible adverse effect on weevil numbers. On the grounds that ibex not only consume large amounts of plants, but often incidentally ingest weevils attached to plants. However, no evidence has been found to date that the presence of weevils affect the food available to ibex [1]. For more information on the ecological context of amensalism interaction, see [2-4] and references cited therein.
象鼻虫与草履虫之间是一种补偿关系，草履虫与山羊以同类植物为食。山羊的存在对草履虫的数量产生了可怕的不利影响。理由是，山羊不仅吃掉大量植物，还经常误食附着在植物上的象鼻虫。然而，迄今为止还没有证据表明象鼻虫的存在会影响山羊的食物[1]。欲了解更多关于同类相互作用的生态背景信息，请参阅 [2-4] 及其中引用的参考文献。

In the last two decades, mathematical modeling has played an important role in understanding amensalism. Sun [5] initially proposed the two-species amensalism model shown below.
在过去二十年中，数学模型在理解同种异形现象方面发挥了重要作用。Sun [5]最初提出了如下所示的双物种同义模型。

where $x(t)$$x(t)$x(t)x(t) and $y(t)$$y(t)$y(t)y(t) denote the densities of the inhibited and unaffected species at time $t$$t$tt, respectively; $r_1$$r_1$r_(1)r_{1} and $r_2$$r_2$r_(2)r_{2} represent the intrinsic growth rates of $x$$x$xx and $y$$y$yy, respectively; $k_1$$k_1$k_(1)k_{1} and $k_2$$k_2$k_(2)k_{2} are the environmental capacities of $x$$x$xx and $y$$y$yy, respectively; $c>0$$c>0$c > 0c>0 reflects the impact exerted by the unaffected species on the inhibited species. The author studied the stability properties of all possible equilibria of (1.1). Since then, to better understand amensalism, (1.1) has been modified to include factors such as non-selective harvesting [6], different inhibition terms [7-9], Allee effect [10-12], and so on. For more achievements on amensalism, one can refer to [13-16].
其中， $x(t)$$x(t)$x(t)x(t) 和 $y(t)$$y(t)$y(t)y(t) 分别表示受抑制物种和不受影响物种在 $t$$t$tt 时间的密度； $r_1$$r_1$r_(1)r_{1} 和 $r_2$$r_2$r_(2)r_{2} 分别表示 $x$$x$xx 和 $y$$y$yy 的内在增长率； $k_1$$k_1$k_(1)k_{1} 和 $k_2$$k_2$k_(2)k_{2} 分别代表 $x$$x$xx 和 $y$$y$yy 的环境容量； $c>0$$c>0$c > 0c>0 反映了未受影响物种对受抑制物种的影响。作者研究了 (1.1) 所有可能平衡的稳定性。此后，为了更好地理解补偿作用，人们对（1.1）进行了修改，加入了非选择性收获[6]、不同抑制项[7-9]、Allee效应[10-12]等因素。关于补偿作用的更多成果，可参阅文献[13-16]。

As for functional response in predation, many scholars believe it is more reasonable to describe the inhibition in amensalism with nonlinear terms like nonlinear functional responses. Beddington-DeAngelis functional response introduced by Beddington [17] and DeAngelis et al. [18] is formed like $\frac{cx}{1+mx+ny}$$\frac{cx}{1+mx+ny}$(cx)/(1+mx+ny)\frac{c x}{1+m x+n y}, which is similar to the well-known Holling type II functional response $\frac{cx}{1+mx}$$\frac{cx}{1+mx}$(cx)/(1+mx)\frac{c x}{1+m x}, but the additional term $ny$$ny$nyn y in the denominator is described as mutual interference between predators. Outstanding statistical evidence from 19 predator-prey systems demonstrates that the BeddingtonDeAngelis functional response provides a better description of predation across the range of predator-prey abundance, suggesting that the Beddington-DeAngelis type functional response performs better than other functional responses [19]. In [7], Guan and Chen presented an amensalism model with Beddington-DeAngelis-like inhibition as follows:
至于捕食中的功能反应，许多学者认为用非线性功能反应等非线性术语来描述同义词中的抑制作用更为合理。Beddington [17] 和 DeAngelis 等人[18] 提出的 Beddington-DeAngelis 功能响应的形式类似于 $\frac{cx}{1+mx+ny}$$\frac{cx}{1+mx+ny}$(cx)/(1+mx+ny)\frac{c x}{1+m x+n y} ，它与著名的霍林 II 型功能响应 $\frac{cx}{1+mx}$$\frac{cx}{1+mx}$(cx)/(1+mx)\frac{c x}{1+m x} 相似，但分母中的附加项 $ny$$ny$nyn y 被描述为捕食者之间的相互干扰。来自 19 个捕食者-被捕食者系统的突出统计证据表明，贝丁顿-德安吉利斯功能响应能更好地描述捕食者-被捕食者丰度范围内的捕食情况，这表明贝丁顿-德安吉利斯型功能响应比其他功能响应表现得更好[19]。在文献[7]中，Guan 和 Chen 提出了一个具有类似贝丁顿-德安吉利斯抑制作用的补偿模型如下：

They looked at the presence and stability of equilibria. Under some conditions, there is the bistability phenomenon. They also analyzed saddle-node and transcritical bifurcations.
他们研究了平衡的存在和稳定性。在某些条件下，存在双稳态现象。他们还分析了鞍节点和跨临界分岔。

Recently, many researchers have explored the influence of Allee effect on species dynamics. Allee effect denotes a favourable relationship between individual fitness
最近，许多研究人员探讨了阿利效应对物种动态的影响。阿利尔效应指的是个体适应性与物种动态之间的有利关系。
and population density. Its mechanisms are the result of species-wide collaboration or facilitation. Better mate finding, environmental conditioning, and collective defence against predators are examples of cooperative behaviours. Existing research indicates that Allee effect can result in more complex dynamics than models without Allee effect. Consequently, Guan and Chen [7] incorporated Allee effect into the unaffected species and obtained the modified system of (1.2),
和种群密度。其机制是整个物种协作或促进的结果。更好地寻找配偶、环境调节和集体防御捕食者就是合作行为的例子。现有研究表明，与没有阿利尔效应的模型相比，阿利尔效应会导致更复杂的动态变化。因此，Guan 和 Chen[7]将阿利效应纳入未受影响的物种，得到了（1.2）的修正系统、

where $\frac{y}{\mu +y}$$\frac{y}{\mu +y}$(y)/(mu+y)\frac{y}{\mu+y} represents the Allee effect and $\mu >0$$\mu >0$mu > 0\mu>0 characterises the intensity of the Allee effect. Although the Allee effect does not affect the population density of both species at a steady-state solution, the solutions of (1.3) tend to its stable steady-state solution significantly slower than those of (1.2) do.
其中， $\frac{y}{\mu +y}$$\frac{y}{\mu +y}$(y)/(mu+y)\frac{y}{\mu+y} 表示阿利效应， $\mu >0$$\mu >0$mu > 0\mu>0 表示阿利效应的强度。虽然在稳态解中，阿利效应并不影响两个物种的种群密度，但 (1.3) 的解比 (1.2) 的解更缓慢地趋向于其稳定的稳态解。

The discrete models are more realistic than the continuous analogs since the statistics of biological sample data are compiled from given time intervals rather than continuously. Furthermore, discrete-time models can be helpful computational models for numerical simulations. In recent years, numerous researchers have explored different ways of incorporating Allee effect into discrete-time population models. For example, the discrete predator-prey model with Holling type II functional response and strong Allee effect was investigated by Zhang and Zou [20]. They created stability and bifurcation conditions for distinct equilibrium points and used feedback control to regulate the chaos in the system. Işık [21] proposed a modified commensal symbiosis model with Allee effect. It is found that the discrete-time model displays diverse and rich dynamical behaviours in the stability properties. The research results of Pal [22] and his colleagues reveal that the Allee effect can stabilize the discrete-time model with Holling IV type functional response. Chen et al. [23] studied a discrete-time model in which prey species have an Allee effect and predator species have other food resources. Their work shows that the Allee effect decreases the population density of prey species at a steady state. When the Allee effect constant grows in the low-value range, it will take longer to achieve a steady state. However, little research has been conducted on discrete-time amensalism models including the Allee effect. Due to its biological and mathematical significance, it is vital to explore the bifurcation analysis of these models. This work aims to contribute in this regard. The model to be studied is obtained by discretizing (1.3), where we allow $\theta$$\theta$theta\theta to be zero.
离散模型比连续模型更真实，因为生物样本数据的统计是根据给定的时间间隔而不是连续的时间间隔编制的。此外，离散时间模型可以作为数值模拟的有用计算模型。近年来，许多研究人员探索了将阿利效应纳入离散时间种群模型的不同方法。例如，Zhang 和 Zou [20] 研究了具有 Holling II 型功能响应和强阿利效应的离散捕食者-猎物模型。他们为不同的平衡点创造了稳定性和分岔条件，并利用反馈控制来调节系统中的混乱。Işık[21]提出了一个改进的具有阿利效应的共生共栖模型。研究发现，该离散时间模型在稳定性方面表现出多样而丰富的动力学行为。Pal[22]及其同事的研究结果表明，阿利效应可以稳定具有霍林 IV 型功能响应的离散时间模型。Chen 等[23]研究了一个离散时间模型，在该模型中，猎物物种具有阿利效应，而捕食者物种具有其他食物资源。他们的研究表明，在稳定状态下，阿利效应会降低猎物物种的种群密度。当阿利效应常数在低值范围内增长时，需要更长的时间才能达到稳定状态。然而，对包括阿利效应在内的离散时间补偿模型的研究还很少。由于其生物学和数学意义，探索这些模型的分岔分析至关重要。本研究旨在对此做出贡献。要研究的模型是通过离散化（1.3）得到的，其中我们允许 $\theta$$\theta$theta\theta 为零。

First, with the transform $x=\frac{\overline{x}}{m},y=\frac{\overline{y}}{n},t=\frac{n\overline{t}}{b_2}$$x=\frac{\overline{x}}{m},y=\frac{\overline{y}}{n},t=\frac{n\overline{t}}{b_2}$x=(( bar(x)))/(m),y=(( bar(y)))/(n),t=(n( bar(t)))/(b_(2))x=\frac{\bar{x}}{m}, y=\frac{\bar{y}}{n}, t=\frac{n \bar{t}}{b_{2}}, after removing the bars, system (1.3) changes
首先，通过变换 $x=\frac{\overline{x}}{m},y=\frac{\overline{y}}{n},t=\frac{n\overline{t}}{b_2}$$x=\frac{\overline{x}}{m},y=\frac{\overline{y}}{n},t=\frac{n\overline{t}}{b_2}$x=(( bar(x)))/(m),y=(( bar(y)))/(n),t=(n( bar(t)))/(b_(2))x=\frac{\bar{x}}{m}, y=\frac{\bar{y}}{n}, t=\frac{n \bar{t}}{b_{2}} ，移除条形图后，系统 (1.3) 将发生变化

where $\alpha =\frac{a_{1}n}{b_2},\beta =\frac{b_{1}n}{b_{2}m},\gamma =\frac{a_{2}n}{b_2},c=\frac{c_1}{b_2}$$\alpha =\frac{a_{1}n}{b_2},\beta =\frac{b_{1}n}{b_{2}m},\gamma =\frac{a_{2}n}{b_2},c=\frac{c_1}{b_2}$alpha=(a_(1)n)/(b_(2)),beta=(b_(1)n)/(b_(2)m),gamma=(a_(2)n)/(b_(2)),c=(c_(1))/(b_(2))\alpha=\frac{a_{1} n}{b_{2}}, \beta=\frac{b_{1} n}{b_{2} m}, \gamma=\frac{a_{2} n}{b_{2}}, c=\frac{c_{1}}{b_{2}}, and $\theta =\mu n$$\theta =\mu n$theta=mu n\theta=\mu n. With the assistance of the piecewise constant parameter method introduced by Jiang and Rogers [24], we convert the model (1.4) (equivalently (1.3)) into a discrete one,
其中 $\alpha =\frac{a_{1}n}{b_2},\beta =\frac{b_{1}n}{b_{2}m},\gamma =\frac{a_{2}n}{b_2},c=\frac{c_1}{b_2}$$\alpha =\frac{a_{1}n}{b_2},\beta =\frac{b_{1}n}{b_{2}m},\gamma =\frac{a_{2}n}{b_2},c=\frac{c_1}{b_2}$alpha=(a_(1)n)/(b_(2)),beta=(b_(1)n)/(b_(2)m),gamma=(a_(2)n)/(b_(2)),c=(c_(1))/(b_(2))\alpha=\frac{a_{1} n}{b_{2}}, \beta=\frac{b_{1} n}{b_{2} m}, \gamma=\frac{a_{2} n}{b_{2}}, c=\frac{c_{1}}{b_{2}} 和 $\theta =\mu n$$\theta =\mu n$theta=mu n\theta=\mu n 。在 Jiang 和 Rogers [24] 提出的片断常数参数法的帮助下，我们将模型 (1.4) （等价于 (1.3)）转换为离散模型、

where $0\le n\le t<n+1$$0\le n\le t<n+1$0 <= n <= t < n+10 \leq n \leq t<n+1 and $[t]$$[t]$[t][t] is the greatest integer less than or equal to $t$$t$tt. System (1.5) is simply solved by direct integration over a unit length time interval. Because the right hand side of system (1.5) is constant over the interval $[n,n+1$$[n,n+1$[n,n+1[n, n+1 ), integrating over $[n,t)$$[n,t)$[n,t)[n, t) and allowing $t\to n+1$$t\to n+1$t rarr n+1t \rightarrow n+1 yield
其中 $0\le n\le t<n+1$$0\le n\le t<n+1$0 <= n <= t < n+10 \leq n \leq t<n+1 和 $[t]$$[t]$[t][t] 是小于或等于 $t$$t$tt 的最大整数。系统 (1.5) 只需在单位长度时间间隔内直接积分即可求解。由于系统 (1.5) 的右边在时间间隔 $[n,n+1$$[n,n+1$[n,n+1[n, n+1 上是常数，因此对 $[n,t)$$[n,t)$[n,t)[n, t) 进行积分并允许 $t\to n+1$$t\to n+1$t rarr n+1t \rightarrow n+1 得到

for $n=0,1,2,\dots$$n=0,1,2,\dots$n=0,1,2,dotsn=0,1,2, \ldots Denoting $x(n)$$x(n)$x(n)x(n) by $x_n$$x_n$x_(n)x_{n} and $y(n)$$y(n)$y(n)y(n) by $y_n$$y_n$y_(n)y_{n}, we thus have obtained the following discrete-time model,
将 $x(n)$$x(n)$x(n)x(n) 表示为 $x_n$$x_n$x_(n)x_{n} ，将 $y(n)$$y(n)$y(n)y(n) 表示为 $y_n$$y_n$y_(n)y_{n} ，我们就得到了下面的离散时间模型、

Obviously, the initial conditions $(x_0,y_0)$$\left(x_0,y_0\right)$(x_(0),y_(0))\left(x_{0}, y_{0}\right) for (1.6) are in ${\mathbb{R}}_{+}^2$${\mathbb{R}}_{+}^2$R_(+)^(2)\mathbb{R}_{+}^{2}. It should be pointed out that the model (1.6) would be used to investigate the interspecific relationship between weevil and Spanish ibex. Here, ibex has a detrimental impact on weevils, which is subject to the Beddington-DeAngelis type functional response. Besides, we further introduced the Allee effect into the ibex species.
显然，（1.6）的初始条件 $(x_0,y_0)$$\left(x_0,y_0\right)$(x_(0),y_(0))\left(x_{0}, y_{0}\right) 在 ${\mathbb{R}}_{+}^2$${\mathbb{R}}_{+}^2$R_(+)^(2)\mathbb{R}_{+}^{2} 中。需要指出的是，（1.6）模型将用于研究草履虫与西班牙山羊之间的种间关系。在这里，西班牙羱羊对草履虫有不利影响，这受到贝丁顿-德安吉利斯（Beddington-DeAngelis）型功能响应的影响。此外，我们还进一步将阿利效应引入到山羊物种中。

The remainder of the paper is structured as follows. In Sect. 2, We address the existence and local stability of all possible fixed points. Then, Sect. 3 provides a comprehensive bifurcation analysis that includes transcritical, pitchfork, fold, and flip bifurcations. Section 4 is devoted to chaos control. Numerical simulations are often used to back up the findings. A brief summary and discussion conclude the study.
本文其余部分的结构如下。在第 2 节中，我们将讨论所有可能定点的存在性和局部稳定性。然后，第3 节提供全面的分岔分析，包括跨临界、叉形、折叠和翻转分岔。第 4 节专门讨论混沌控制。研究中经常使用数值模拟来支持研究结果。本研究最后进行了简要总结和讨论。

A fixed point $(x,y)$$(x,y)$(x,y)(x, y) of (1.6) satisfies
(1.6) 的定点 $(x,y)$$(x,y)$(x,y)(x, y) 满足