Invasion by an exotic species in a three-species competition-diffusion system
<p>We consider the situation where an exotic species <em>w</em> invades an ecosystem inhabited by two native species <em>u</em> and <em>v</em>. All species are competing for the same limited resource. Supposing that <em>u</em> and <em>v</em> are not able to coexist in the absence of the invader, we want to determine whether a successful invasion by <em>w</em> may allow all species to coexist (competitor-mediated coexistence). Mathematically, this problem can be modelled by the following three-species competition-diffusion system<br> <span class="math-tex">\( \left\{ \begin{alignedat}{6} u_t &= d_1 \, \Delta u &&+ (r_1 &&- u &&- b_{12} \, v &&- b_{13} \, w &&)\,u, \\ v_t &= d_2 \, \Delta v &&+ (r_2 &&- v &&- b_{21} \, u &&- b_{23} \, w &&)\,v, \\ w_t &= d_3 \, \Delta w &&+ (r_3 &&- w &&- b_{31} \, u &&- b_{32} \, v &&)\,w, \end{alignedat} \right.\)</span><br> where all parameters are positive constants.</p> <p>We are interested in the case in which the invading species is weaker than the native ones, i.e., it is not able to survive in the diffusion-free system obtained by setting <em>d</em><sub>1</sub> = <em>d</em><sub>2</sub> = <em>d</em><sub>3</sub> = 0.<br> We fix all parameters as<br> <span class="math-tex">\( \begin{aligned} & d_1 = d_2 = d_3 = 1, \\ & r_1 = r_2 = 28, \\ & \begin{aligned} b_{12} &= 22/21, & b_{13} &= 4, \\ b_{21} &= 1.87, & b_{23} &= 3/4, \\ b_{31} &= 26/21, & b_{32} &= 22/21, \\ \end{aligned} \end{aligned}\)</span><br> and leave <em>r</em><sub>3</sub>, which measures the strength of the exotic species, as a free parameter. Depending on the value of <em>r</em><sub>3</sub>, the invasion can be either successful or not and competitor-mediated coexistence may or may not occur, as can be seen in the movies here presented. The species <em>u</em>, <em>v</em> and <em>w</em> are denoted by the red, green and blue colours respectively. The yellow line marks the interface between the species <em>u</em> and <em>v</em>. We remark that competitor-mediated coexistence only occurs for intermediate values of <em>r</em><sub>3</sub>.</p>
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32/100
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- Harmonization
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- Reuse readiness
- 8
- Engagement
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