# American Institute of Mathematical Sciences

## The impact of toxins on competition dynamics of three species in a polluted aquatic environment

 1 School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, China 2 School of Mathematical and Statistical Sciences, Southwest University, Chongqing 400715, China 3 School of Mathematical and Statistical Sciences, Hubei University of Science and Technology, Xianning 437100, China

*Corresponding author: Qihua Huang

Received  February 2020 Revised  May 2020 Published  July 2020

Fund Project: The second author is supported by the NSF of China (11871235). The third author is supported by the NSF of China (11871060), the Venture and Innovation Support Program for Chongqing Overseas Returnees (7820100158), the Fundamental Research Funds for the Central Universities (XDJK2018B031), and the faculty startup fund from Southwest University (20710948)

Accurately assessing the risks of toxins in polluted ecosystems and finding factors that determine population persistence and extirpation are important from both environmental and conservation perspectives. In this paper, we develop and study a toxin-mediated competition model for three species that live in the same polluted aquatic environment and compete for the same resources. Analytical analysis of positive invariance, existence and stability of equilibria, sensitivity of equilibria to toxin are presented. Bifurcation analysis is used to understand how the environmental toxins, plus distinct vulnerabilities of three species to toxins, affect the competition outcomes. Our results reveal that while high concentrations lead to extirpation of all species, sublethal levels of toxins affect competition outcomes in many counterintuitive ways, which include boosting coexistence of species by reducing the abundance of the predominant species, inducing many different types of bistability and even tristability, generating and reducing population oscillations, and exchanging roles of winner and loser in competition. The findings in this work provide a sound theoretical foundation for understanding and assessing population or community effects of toxicity.

Citation: Yuyue Zhang, Jicai Huang, Qihua Huang. The impact of toxins on competition dynamics of three species in a polluted aquatic environment. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020219
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The illustration of Theorem 3.7
Bifurcation diagrams with respect to toxin level $T$ for the case where species-2 only equilibrium $E_2$ is globally asymptotically stable when $T = 0$. Parameters: $b_1 = 1, \ b_2 = 1.06, \ b_3 = 1.13, \ p_1 = p_2 = p_3 = k_1 = k_2 = k_3 = 1, \ m_1 = 0.44, \ m_2 = 0.5, \ m_3 = 0.56, \ c_{12} = 1.13, \ c_{13} = 1.19, \ c_{21} = 1.06, \ c_{23} = 0.88, \ c_{31} = 0.75, \ c_{32} = 1.24$. Here $T_1^* = 0.28$, $T_2^*\approx0.27$, $T_3^*\approx0.27$
(a) The Hopf bifurcation diagram corresponding to Fig. 2; (b) Limit cycle for $T = 0.18069$ in (a); (c) The time series diagram corresponding to (b); (d) The time series diagram of heteroclinic loop for $T = 0.1801$; (e) The phase portrait corresponding to (d). The other parameters are the same as those in Fig. 2
Bifurcation diagrams with respect to toxin level $T$ for the case where both species 2-only equilibrium $E_2$ and species 3-only equilibrium $E_3$ are locally asymptotically stable when $T = 0$. Parameters: $c_{13} = 1.15, \ c_{23} = 1.19, \ c_{32} = 1.13$, the other parameters and the values of $T_i^*\ (i = 1, \ 2, \ 3)$ are the same as those in Fig. 2
Bifurcation diagrams with respect to toxin level $T$ for the case where the coexistence equilibrium $E^*$ is globally asymptotically stable when $T = 0$. Part of panel (a) is enlarged by panel (b) and part of panel (c) is enlarged by panel (d). Parameters: $b_1 = 1, \ b_2 = 0.97, \ b_3 = 0.96, \ p_1 = p_2 = p_3 = k_1 = k_2 = k_3 = 1, \ m_1 = 0.52, \ m_2 = 0.47, \ m_3 = 0.53, \ c_{12} = 0.11, \ c_{13} = 0.16, \ c_{21} = 0.16, \ c_{23} = 0.21, \ c_{31} = 0.15, \ c_{32} = 0.22$. Here $T_1^* = 0.24$, $T_2^*\approx0.25$, $T_3^*\approx0.22$
Bifurcation diagrams with respect to toxin level $T$, where the parameters are the same to those in Fig. 2 except that $p_2 = 0.5, \ p_3 = 0.2$. Here, $T_1^* = 0.28$, $T_2^*\approx 0.54$, $T_3^*\approx 1.34$
Bifurcation diagrams with respect to toxin level $T$, where the parameters are similar to those in Fig. 2 except that $p_2 = 2, \ p_3 = 2.5$. Here $T_1^* = 0.28$, $T_2^*\approx0.14$, $T_3^*\approx 0.11$
The asymptotic dynamics on $\Sigma$ of system (8). $\bullet$ signifies an attractive equilibrium on $\Sigma$, $\circ$ signifies a repellent equilibrium on $\Sigma$, the intersection of its hyperbolic manifolds signifies a saddle on $\Sigma$
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