A stochastic predator–prey model with two competitive preys and Ornstein–Uhlenbeck process

Figures & data

Table 1. List of parameter values of system (5(5) $\{\begin{array}{l}\mathrm{d}{x}_1(t)=x_1(t)[a_1(t)-b_{11}{x}_1(t)-b_{12}{x}_2(t)-b_{13}{x}_3(t)]\mathrm{d}t,\\ \mathrm{d}{x}_2(t)=x_2(t)[a_2(t)-b_{21}{x}_1(t)-b_{22}{x}_2(t)-b_{23}{x}_3(t)]\mathrm{d}t,\\ \mathrm{d}{x}_3(t)=x_3(t)[-a_3(t)+b_{31}{x}_1(t)+b_{32}{x}_2(t)-b_{33}{x}_3(t)]\mathrm{d}t,\\ \mathrm{d}{a}_1(t)={\alpha }_1[{\overline{a}}_1-a_1(t)]\mathrm{d}t+{\sigma }_1\mathrm{d}{B}_1(t),\\ \mathrm{d}{a}_2(t)={\alpha }_2[{\overline{a}}_2-a_2(t)]\mathrm{d}t+{\sigma }_2\mathrm{d}{B}_2(t),\\ \mathrm{d}{a}_3(t)={\alpha }_3[{\overline{a}}_3-a_3(t)]\mathrm{d}t+{\sigma }_3\mathrm{d}{B}_3(t).\end{array}$(5) ).

Parameters	Values	Source
${\overline{a}}_1$	0.8	[17, 28]
${\overline{a}}_2$	0.6	[17, 28]
${\overline{a}}_3$	0.05	[17, 18, 28]
$b_{11}$	1.0	[17, 18, 28]
$b_{12}$	0.27	[17, 28]
$b_{13}$	0.4	[17, 28]
$b_{21}$	0.1	[17, 28]
$b_{22}$	2.0	[17, 18, 28]
$b_{23}$	0.2	[17, 28]
$b_{31}$	0.4	[17, 28]
$b_{32}$	0.1	[17, 28]
$b_{33}$	3.0	[17, 18, 28]
${\alpha }_1$	0.2	Estimated
${\alpha }_2$	0.1	Estimated
${\alpha }_3$	0.2	Estimated

Figure 1. The left column displays the trajectory of the species $x_1$, $x_2$ and $x_3$ of system (5(5) $\{\begin{array}{l}\mathrm{d}{x}_1(t)=x_1(t)[a_1(t)-b_{11}{x}_1(t)-b_{12}{x}_2(t)-b_{13}{x}_3(t)]\mathrm{d}t,\\ \mathrm{d}{x}_2(t)=x_2(t)[a_2(t)-b_{21}{x}_1(t)-b_{22}{x}_2(t)-b_{23}{x}_3(t)]\mathrm{d}t,\\ \mathrm{d}{x}_3(t)=x_3(t)[-a_3(t)+b_{31}{x}_1(t)+b_{32}{x}_2(t)-b_{33}{x}_3(t)]\mathrm{d}t,\\ \mathrm{d}{a}_1(t)={\alpha }_1[{\overline{a}}_1-a_1(t)]\mathrm{d}t+{\sigma }_1\mathrm{d}{B}_1(t),\\ \mathrm{d}{a}_2(t)={\alpha }_2[{\overline{a}}_2-a_2(t)]\mathrm{d}t+{\sigma }_2\mathrm{d}{B}_2(t),\\ \mathrm{d}{a}_3(t)={\alpha }_3[{\overline{a}}_3-a_3(t)]\mathrm{d}t+{\sigma }_3\mathrm{d}{B}_3(t).\end{array}$(5) ) and their corresponding deterministic model (1(1) $\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{x}_1(t)}{\mathrm{d}t}=x_1(t)[a_1-b_{11}{x}_1(t)-b_{12}{x}_2(t)-b_{13}{x}_3(t)],{\displaystyle \frac{\mathrm{d}{x}_2(t)}{\mathrm{d}t}=x_2(t)[a_2-b_{21}{x}_1(t)-b_{22}{x}_2(t)-b_{23}{x}_3(t)],{\displaystyle \frac{\mathrm{d}{x}_3(t)}{\mathrm{d}t}=x_3(t)[-a_3+b_{31}{x}_1(t)+b_{32}{x}_2(t)-b_{33}{x}_3(t)],}}}\end{array}$(1) ) with noise intensities ${\sigma }_1={\sigma }_2={\sigma }_3=0.001$. The right column shows the frequency histograms and marginal density functions of $x_1$, $x_2$ and $x_3$ in system (5(5) $\{\begin{array}{l}\mathrm{d}{x}_1(t)=x_1(t)[a_1(t)-b_{11}{x}_1(t)-b_{12}{x}_2(t)-b_{13}{x}_3(t)]\mathrm{d}t,\\ \mathrm{d}{x}_2(t)=x_2(t)[a_2(t)-b_{21}{x}_1(t)-b_{22}{x}_2(t)-b_{23}{x}_3(t)]\mathrm{d}t,\\ \mathrm{d}{x}_3(t)=x_3(t)[-a_3(t)+b_{31}{x}_1(t)+b_{32}{x}_2(t)-b_{33}{x}_3(t)]\mathrm{d}t,\\ \mathrm{d}{a}_1(t)={\alpha }_1[{\overline{a}}_1-a_1(t)]\mathrm{d}t+{\sigma }_1\mathrm{d}{B}_1(t),\\ \mathrm{d}{a}_2(t)={\alpha }_2[{\overline{a}}_2-a_2(t)]\mathrm{d}t+{\sigma }_2\mathrm{d}{B}_2(t),\\ \mathrm{d}{a}_3(t)={\alpha }_3[{\overline{a}}_3-a_3(t)]\mathrm{d}t+{\sigma }_3\mathrm{d}{B}_3(t).\end{array}$(5) ).

Figure 2. Numerical simulations for: (i) the frequency histogram fitting density curves of $x_1$, $x_2$, $x_3$, $a_1$, $a_2$ and $a_3$ of system (5(5) $\{\begin{array}{l}\mathrm{d}{x}_1(t)=x_1(t)[a_1(t)-b_{11}{x}_1(t)-b_{12}{x}_2(t)-b_{13}{x}_3(t)]\mathrm{d}t,\\ \mathrm{d}{x}_2(t)=x_2(t)[a_2(t)-b_{21}{x}_1(t)-b_{22}{x}_2(t)-b_{23}{x}_3(t)]\mathrm{d}t,\\ \mathrm{d}{x}_3(t)=x_3(t)[-a_3(t)+b_{31}{x}_1(t)+b_{32}{x}_2(t)-b_{33}{x}_3(t)]\mathrm{d}t,\\ \mathrm{d}{a}_1(t)={\alpha }_1[{\overline{a}}_1-a_1(t)]\mathrm{d}t+{\sigma }_1\mathrm{d}{B}_1(t),\\ \mathrm{d}{a}_2(t)={\alpha }_2[{\overline{a}}_2-a_2(t)]\mathrm{d}t+{\sigma }_2\mathrm{d}{B}_2(t),\\ \mathrm{d}{a}_3(t)={\alpha }_3[{\overline{a}}_3-a_3(t)]\mathrm{d}t+{\sigma }_3\mathrm{d}{B}_3(t).\end{array}$(5) ) with 200,000 iteration points, respectively. (ii) The marginal probability densities of $x_1$, $x_2$, $x_3$, $a_1$, $a_2$ and $a_3$ of system (5(5) $\{\begin{array}{l}\mathrm{d}{x}_1(t)=x_1(t)[a_1(t)-b_{11}{x}_1(t)-b_{12}{x}_2(t)-b_{13}{x}_3(t)]\mathrm{d}t,\\ \mathrm{d}{x}_2(t)=x_2(t)[a_2(t)-b_{21}{x}_1(t)-b_{22}{x}_2(t)-b_{23}{x}_3(t)]\mathrm{d}t,\\ \mathrm{d}{x}_3(t)=x_3(t)[-a_3(t)+b_{31}{x}_1(t)+b_{32}{x}_2(t)-b_{33}{x}_3(t)]\mathrm{d}t,\\ \mathrm{d}{a}_1(t)={\alpha }_1[{\overline{a}}_1-a_1(t)]\mathrm{d}t+{\sigma }_1\mathrm{d}{B}_1(t),\\ \mathrm{d}{a}_2(t)={\alpha }_2[{\overline{a}}_2-a_2(t)]\mathrm{d}t+{\sigma }_2\mathrm{d}{B}_2(t),\\ \mathrm{d}{a}_3(t)={\alpha }_3[{\overline{a}}_3-a_3(t)]\mathrm{d}t+{\sigma }_3\mathrm{d}{B}_3(t).\end{array}$(5) ). All of the parameter values are the same as in Figure 1.

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Availability of data and materials

Data sharing is not applicable to this article as no datasets are generated or analysed during the current study.