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Only mimic-species can survive when external noises are added in a ratio-dependent community dynamics model

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Abstract

In this paper, we present a ratio-dependent community dynamics model on Batesian mimicry. In ecology, Batesian mimicry is very important which focus mainly on the patterns in model-species and the mimic-species. Here, we study a stochastically forced basic model-mimic community dynamics model. Without any environmental noise, both model-species and mimic-species can survive. However, external noises push the model-species to extinct and only mimic-species can stable. We discuss a mechanism of the noise-induced transition in the basin of attraction of a stable interior equilibrium point. We apply the stochastic sensitivity function technique to construct a confidence domain near a stable interior equilibrium point and study different noise intensity for the transition.

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Acknowledgements

Koushik Garain is supported by University Grant Commission, India (Student Id: MAY2018-442750). I am very grateful to Dr. Partha Sarathi Mandal for advice and helpful discussion.

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Appendix 1

Appendix 1

Stochastic sensitivity function (SSF) technique: We can represent equation (4) for the system of stochastic differential equations

$$\begin{aligned} {\dot{x}}=f(x)+\varepsilon \sigma (x)\xi , \end{aligned}$$
(A.1)

where x is n-vector, f(x) is a n-vector function, \(\xi (t)\) is a n-dimensional Gaussian white noise satisfying \(\langle \xi (t)\rangle =0,\langle \xi (t)\xi (\tau ) \rangle =\delta (t-\tau )I\), I is an identity matrix, and \(\sigma (x)\) is \(n\times n\) matrix-valued function of disturbances with intensity \(\varepsilon \).

We must find a stable interior equilibrium point of the deterministic system (2)–(3). Let the stable equilibrium point is \(x_*\). When we add noise in the system, ordinary differential equation is transformed to a stochastic differential equation or Ito equation. It is supposed that attractor \(x_*\) is exponentially stable. It means that for the small neighborhood N of the attractor \(x_*\), there exist constants \(M > 0\); \(k > 0\), such that for any solution x(t) of the deterministic system with \(x(0) = x_0 \in N\), the following inequality holds:

$$\begin{aligned} \parallel \bigtriangleup (x(t))\parallel \le M e^{-kt}\parallel \bigtriangleup (x_0)\parallel . \end{aligned}$$

Here, \(\bigtriangleup (x) = x-x_*\) is a vector of a deviation of the point x from the attractor.

Now, for the corresponding stochastic system, a set of random trajectories is generated. For small noise, the random trajectories are localized in the neighborhood of \(x_*\) owing to the exponential stability of \(x_*\) and form a stationary probabilistic distribution \(\rho (x,\varepsilon )\), which satisfies the following stationary Fokker–Planck–Kolmogorov equation:

$$\begin{aligned} \frac{\varepsilon ^2}{2}\sum \frac{\partial ^2}{\partial x_{i}\partial x_{j}}(a_{ij}\rho )-\sum \frac{\partial }{\partial x_{i}}(f_{i}\rho )=0,a_{ij}=[\sigma \sigma ^T]_{ij}.\nonumber \\ \end{aligned}$$
(A.2)

Even for a two-dimensional case, it is very difficult to solve this equation. However, we can find solution by approximation. For a small noise, we use the approximation of \(\rho (x,\varepsilon )\), based on the quasipotential v(x) [39]

$$\begin{aligned} \rho (x,\varepsilon ) \thickapprox K exp\left( -\frac{v(x)}{\varepsilon ^2}\right) , \end{aligned}$$

where v(x) satisfies the Hamilton–Jacobi equation

$$\begin{aligned} \left( f,\frac{\partial v}{\partial x}\right) +\frac{1}{2}\left( \frac{\partial v}{\partial x},\sigma \sigma ^{T}\frac{\partial v}{\partial x}\right) =0. \end{aligned}$$

Quadratic approximation of v(x) near the stable equilibrium point is \(v(x) \thickapprox \frac{1}{2}(x-x_*,V(x-x_*)),\) since \(v(x_*)=0\) and \(\frac{\partial v}{\partial x}\big |_{x_*}=0\). V is a positive definite \(n\times n\) matrix. Now, the Gaussian approximation of \(\rho (x,\varepsilon )\) is

$$\begin{aligned} \rho (x,\varepsilon ) \thickapprox Kexp\left( -\frac{(x-x_*,W^{-1}(x-x_*))}{2\varepsilon ^2}\right) , \end{aligned}$$

with the covariance matrix \(\varepsilon ^2W\), where \(W=V^{-1}\).

A dispersion (mean-square deviation) of random states near \(x_*\) can be approximated by the following formula [40]:

$$\begin{aligned} E(x-x_*)(x-x_*)^T \thickapprox \varepsilon ^2W. \end{aligned}$$
(A.3)

Here, the matrix W is positive definite and it is the solution of the following matrix equation [40]:

$$\begin{aligned} FW+WF^T+S=0, F=\frac{\partial f}{\partial x}(x_*),S=\sigma (x_*)\sigma ^T(x_*).\nonumber \\ \end{aligned}$$
(A.4)

The matrix W is called the stochastic sensitivity function and it characterizes a configurational arrangement of random states of the stochastic system around the deterministic equilibrium \(x_*\):

$$\begin{aligned} (x-x_*,W^{-1}(x-x_*))=2\varepsilon ^2k. \end{aligned}$$

Here, \(k=-\ln (1-P)\), P is a fiducial probability. This means that the random states belong to the interior of this ellipse with probability P. Let \(\lambda _{1},\lambda _{2}\) are eigenvalues, and \(v_{1},v_{2}\) are corresponding normalized eigenvectors of the matrix W, respectively. For the coordinates \(z_{1}=(x-x_*,v_{1})\) and \(z_{2}=(x-x_*,v_{2})\), the equation of the confidence ellipse in a basis of \(v_1, v_2\) with the origin at the point \(x_*\) can be written in a standard form:

$$\begin{aligned} \frac{z^2_{1}}{\lambda _{1}}+\frac{z^2_{2}}{\lambda _{2}}=2\varepsilon ^2k. \end{aligned}$$

Confidence ellipse is treated as a simple geometrical representation for the description of a configurational arrangement of random states near stable equilibrium.

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Garain, K. Only mimic-species can survive when external noises are added in a ratio-dependent community dynamics model. Eur. Phys. J. Spec. Top. 230, 3381–3387 (2021). https://doi.org/10.1140/epjs/s11734-021-00111-2

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