Formation of stationary periodic patterns in a model of two competing populations with chemotaxis

Abstract

One of the classical models in mathematical biology is the Lotka-Volterra competition model, describing the dynamics of two populations competing for resources. Two possible regimes in this system are given by their coexistence or extinction of a weaker population. In a distributed system with diffusive spatial coupling, travelling fronts occur, corresponding to transitions between stationary states. In this work we will consider the competition model extended by extra interaction between involved populations which is given by chemotactic coupling, namely, assuming that one species produces a chemical agent which causes the taxis of another species. It is known that in a one-species model (i.e. Keller-Segel model) production of chemoattractor results in formation of stationary periodic (or Turing-type) patterns. In this work, we will investigate conditions for the formation of stationary periodic patterns in a two-species competition model with chemotaxis. We show that in this system periodic patterns can emerge in the course of Turing-type instability (classical way) or from a stable steady state, corresponding to the extinction of one of the species, due to a finite, or over-threshold, amplitude disturbance. We study the characteristics of emerging periodic pattern, such as its amplitude and wavelength, by means of Fourier analysis. We also perform computational simulations to verify our analytical results.

Figures (10)

• Figure 1: Stability of the coexistence steady state depending on interspecific competition. (a): Conditions (\ref{['RHcriteria']}) are verified for fixed parameters $D_1=D_2=1$, $r_1=r_2=0.1$, $\chi=-10$ and $k=0.2$, in the region $b_1, b_2 <1$. In domain $\delta_2$, condition for $a_3$ is violated, i.e. $a_3<0$. (b): stationary periodic pattern obtained from numerical simulations by fixing $b_1=b_2=0.7 \in \delta_2$. Solid blue and dotted red lines represent the density of $u$ and $v$, respectively. Dash-dotted cyan line shows the concentration profile of the chemical which almost coincides with that of $v$.
• Figure 2: Most unstable wavenumber $k$ for different chemotactic strengths and competition factors.(a): The wavelength $\lambda$ as function of the wavenumber $k$ for different chemotactic strengths: $\chi=\{-10 \hbox{(solid)}, -50 \hbox{(dashed)}, -90 \hbox{(dotted)}\}.$(b): Minimum value of $b$ that initiates formation of patterns and the most unstable wavenumber $k$ that ensures maximal size of $\delta_2$. Fixed parameters: $D_1=D_2=1$, $r_1=r_2=0.1$, $\chi=-10$ and $b_1=b_2=b$.
• Figure 3: Domains for types of patterns that can form after disturbance of the coexistence state in the weak competition case in model \ref{['twospecieschem']}. Domains for the formation of stationary periodic patterns on the planes (a): ($b,D_1$); (b): ($b,\chi$); (c): ($b,r_1$); (d): ($b,r_2$). Used default set of parameters: $D_1 = D_2 = 1$, $r_1 = r_2 = 0.1$ and $\chi = -10$. Solid lines represent results from numerical simulations and dashed lines represent analytical results.
• Figure 4: Simulation of pattern obtained in the weak-strong competition case when $b_2>1$.(a): An initially large perturbation of the steady state $(1, 0, 0)$ (dash-dotted cyan line). (b): $v$ starts producing a chemical agent $c$ (dotted red line) which repels species $u$ (solid blue line). (c): Density of $u$ slowly starts increasing in the middle, while the density of $v$ and concentration of $c$ slowly start decaying (d): Stationary pattern consisting of two full spikes has formed. Parameter values: $D_1=D_2=1, r_1=r_2=0.1, \chi=-10$, $b_1=0.7, b_2=1.7$, medium size, $L=50$, and the amplitude of perturbation, $\tilde{v}=0.9.$
• Figure 5: The dependence of the minimal perturbation amplitude, $\tilde{v}$, from the steady state $(1, 0, 0)$, required to generate periodic patterns, on the model parameters. Dependence on the diffusion coefficients (panel $(a)$), chemotactic sensitivity (panel $(b)$), competition strength $b_1$ (panel $(c)$), and $b_2$ (panel $(d)$), reproduction rate $r_1$ (panel $(e)$) and $r_2$ (panel $(f)$). Default set of parameters: $D_1=D_2=1$, $\chi=-10$, $r_1=r_2=0.1$, $b_1=0.7$ and $b_2=1.7$.