Phase Transitions in a Logistic Metapopulation Model with Nonlocal Interactions

Abstract

The presence of one or more species at some spatial locations but not others is a central matter in ecology. This phenomenon is related to ecological pattern formation. Nonlocal interactions can be considered as one of the mechanisms causing such a phenomenon. We propose a single-species, continuous time metapopulation model taking nonlocal interactions into account. Discrete probability kernels are used to model these interactions in a patchy environment. A linear stability analysis of the model shows that solutions to this equation exhibit pattern formation if the dispersal rate of the species is sufficiently small and the discrete interaction kernel satisfies certain conditions. We numerically observe that traveling and stationary wave-type patterns arise near critical dispersal rate. We use weakly nonlinear analysis to better understand the behavior of formed patterns. We show that observed patterns arise through both supercritical and subcritical bifurcations from spatially homogeneous steady state. Moreover, we observe that as the dispersal rate decreases, amplitude of the patterns increases. For discontinuous transitions to instability, we also show that there exists a threshold for the amplitude of the initial condition, above which pattern formation is observed.

Appendices

DFS and Linear Analysis

1.1 Calculation of DFS for a Class of Uniform Kernels

Using the periodicity of $\mathbf u^{(r,a)}$ and complex exponentials, one can calculate $U_k$ for any nonzero integer $k\in S$ as follows:

$$\begin{aligned} U_k= & {} \,\,\sum _{n=0}^{r+a}u_n^{(r,a)}e^{-2j\pi nk/N}+ \sum _{n=r+a-N+1}^{-1}u_{(n)_N}^{(r,a)}e^{-2j\pi nk/N}\\= & {} \,\,\frac{1}{2r+1}\sum _{n=-r+a}^{r+a}e^{-2j\pi nk/N}\\= & {} \,\,\frac{e^{-2j\pi n(a-r)/N}}{2r+1}\sum _{l=0}^{2r}e^{-2j\pi nl/N}\\= & {} \,\,\frac{1}{2r+1}\frac{e^{-2j\pi k(a-r)/N}-e^{-2j\pi k(a+r+1)/N}}{1-e^{-2j\pi k/N}}. \end{aligned}$$

Clearly $U_0=1$ from the first equality. We would like to note that the characteristic function of discrete uniform distribution with support $\{a-r,a-r+1,\ldots , r+a\}$ is given by

$$\begin{aligned} \mathcal C(t)=\frac{e^{j(a-r)t}-e^{j(a+r+1)t}}{(2r+1)(1-e^{jt})}. \end{aligned}$$

Hence above given formula can be obtained from the characteristic equation by evaluating it at discrete values $t=-\,\,2\pi k/N$ for $k\in S.$ It is also possible to use other discrete distributions and compute their DFS by using the shift theorem and their characteristic functions (Smith 2007; Mandal and Asif 2007).

1.2 Discussion Regarding Assumption 1

To show Assumption 1 is not redundant, it is enough to construct an example for which the state $\mathbf 1$ is unstable for any $\delta >0.$ Consider a habitat with 8 patches, i.e., $N=8.$ Take the dispersal kernel $\mathbf d$ with $d_2=d_{<-\,\,2>_8}=d_6=\frac{1}{2}.$ The DFS of this vector is given by $\mathbf D=( 1 ,0 ,-\,\,1 ,0 ,1 ,0 ,-\,\,1 ,0).$ Note that $D_4=1.$ Consider also uniform kernel $\mathbf u^{(1,0)}$ as the interaction kernel. One can calculate the DFS of this kernel as $\mathbf C=(1.0000 , 0.8047 , 0.3333 , -\,\,0.1381 , -\,\,0.3333 -\,\,0.1381 , 0.3333 , 0.8047).$ This implies one of the eigenvalues of the coupled ODE system $\lambda (\delta ,4)=0.3333$ and hence the state is unstable for any $\delta >0.$

Derivation of Cubic S–L Equation

Before proceeding the perturbation analysis, define the following complex constants

$$\begin{aligned} \mathcal L_{\delta _0}^n= njw_0-\delta _0\big (D_{<nk_c>_N}-1\big )+C_{<nk_c>} \end{aligned}$$

for $n \in S.$ In addition, note that we need the following technical assumption to be able to obtain cubic S–L equation.

Assumption 2

We assume that $k_c\ne 0$ and $<4k_c>_N\ne 0$ regarding the system parameters $k_c$ and N.

Plugging the solution (12) into (3) gives the following relation at level $O(\varepsilon ):$

$$\begin{aligned} A(\tau )\mathcal L_{\delta _0}^1W_1=0. \end{aligned}$$

Hence at this level one obtains the linear relation (5) and it is not possible to gather information about the amplitude A.

At level $O(\varepsilon ^2)$ we obtain the following equation:

$$\begin{aligned} \mathcal L_{\delta _0}^0A_{20}W_0+\left( \mathcal L_{\delta _0}^2A_{22}W_2+c.c.\right) =\gamma _{20}AA^cW_0+\left( \gamma _{22}A^2W_2 +c.c\right) \end{aligned}$$

where $\gamma _{20}=-\,\,2\mathcal R[C_{<k_c>_N}]$ and $\gamma _{22}=-\,\,C_{<k_c>_N}.$

By Assumption 2, it is easy to see that $W_0,$ and $W_{\pm \,\,2}$ are linearly independent vectors ( $<4k_c>_N\ne 0$ implies $<2k_c>_N\ne 0$). Hence we have :

$$\begin{aligned} A_{20}= & {} \gamma _{20}AA^c\\ A_{22}= & {} \frac{\gamma _{22}}{\mathcal L_{\delta _0}^2}A^2. \end{aligned}$$

Note also that Assumption 2 guarantees that $W_2$ is not in the kernel of the linearized operator so that $\mathcal L_{\delta _0}^2\ne 0.$

At the level $O(\varepsilon ^3)$ we obtain the following equality

$$\begin{aligned} \mathcal L_{\delta _0}^3A_{33}W_3+c.c.=\bigl (-A_\tau (\tau )+\mu \varPhi A+\varPsi A^2A^c \bigr )W_1+ \gamma _{33} A^3+c.c. \end{aligned}$$

(19)

where

$$\begin{aligned} \varPhi= & {} (D_{<k_c>_N}-1), \end{aligned}$$

(20)

$$\begin{aligned} \varPsi= & {} -\gamma _{20}(1+C_{<k_c>_N})-\frac{\gamma _{22}}{\mathcal L_{\delta _0}^2}(C_{<k_c>_N}^c+C_{<2k_c>_N})\nonumber \\ \text { and } \end{aligned}$$

(21)

$$\begin{aligned} \gamma _{33}= & {} -\frac{\gamma _{22}}{\mathcal L_{\delta _0}^2}(C_{<k_c>_N}+C_{<2k_c>_N}). \end{aligned}$$

(22)

By Assumption 2, one can easily see that $<3k_c>_N\ne \pm k_c.$ Hence, we can easily conclude that the coefficient of $W_1$ is zero in (19), from which we get quintic S–L equation (13).

Derivation of Quintic S–L Equation

To obtain quintic S–L equation, we need more restrictions. Here we suppose that Assumption 2 holds in addition to the following restrictions.

Assumption 3

We assume that $<6k_c>_N\ne 0$ regarding the system parameters $k_c$ and N.

Taking into account that the cubic S–L equation (13) is still valid for complex amplitude A, we take the solution at level $O(\varepsilon ^3)$ as follows

$$\begin{aligned} \mathbf p^{(3)}=\gamma _{3} A^3 W_3 +c.c \end{aligned}$$

(23)

where $\gamma _3=\frac{\gamma _{33}}{\mathcal L_{\delta _0}^3}.$ In addition, for the sake of simplicity, we denote $\frac{\gamma _{22}}{\mathcal L_{\delta _0}^2}$ by $\gamma _2.$

Here we consider an expansion of solution $\mathbf p$ as follows:

$$\begin{aligned} \mathbf p=1+\sum _{m=1}^{4}\varepsilon ^m \mathbf p^{(m)} (t,\tau ,\tau _1)+O(\varepsilon ^5) \end{aligned}$$

(24)

where $\mathbf p^{(i)}$ for $i=1,2,3$ are as defined in Sect. 4.1 and the fourth term has the following form:

$$\begin{aligned} \mathbf p^{(4)}=A_{40}W_0+\big (A_{42}W_2+A_{44}W_4 +c.c.\big ). \end{aligned}$$

Plugging this solution into Eq. (3) one obtains $A_{20}$ and $A_{22}$ as given in Appendix B. The solution $\mathbf p^{(3)}$ is given by (23). Hence we need to determine $\mathbf p^{(4)}.$ At levels $O(\varepsilon ^4)W_i$ for $i=,0,2,4$ we get the following equalities:

$$\begin{aligned} A_{40} =\gamma _{40}^4 |A|^4 +\gamma _{40}^2 |A|^2 \text { and } A_{42} =\gamma _{42}^3A| A|^2 +\gamma _{42}^2 A^2 \end{aligned}$$

where

$$\begin{aligned} \gamma _{40}^4= & {} -\,\,\Bigg (\gamma _{20}^2 + \gamma _{20}(\varPsi ^c + \varPsi ) + \gamma _2\gamma _2^c\left( C_{<2k_c>_N} + C_{<2k_c>_N}^c\right) \Bigg )\\ \gamma _{40}^2= & {} -\,\,\mu \gamma _{20} \big (\varPhi + \varPhi ^c\big )\\ \gamma _{42}^3= & {} \,\,-\Bigg (\gamma _3\left( C_{<3k_c>_N} + C_{<k_c>_N}^c\right) + \gamma _2(\gamma _{20}\big (1+ C_{<2k_c>_N}\big )+ 2 \varPsi )\Bigg )/\mathcal L_{\delta _0}^2\\ \gamma _{42}^2= & {} -\,\,\gamma _2 \Bigg (\mu \big (1-D_{<2k_c>_N}\big ) + 2\varPhi \Bigg )/\mathcal L_{\delta _0}^2. \end{aligned}$$

Note that function $A_{44}$ does not contribute at level $O(\varepsilon ^5)W_1.$ Above equalities require the linear independency of vectors $W_0,$ $W_{\pm \,2}$ and $W_\pm \,4$ which follows from Assumptions 2 and 3. Hence, at the level $O(\varepsilon ^5)W_1$, we have the following equation:

$$\begin{aligned} A_{\tau _1} - \mu \varPhi A + \varTheta _3A| A|^2 + \varTheta _5A| A|^4=0 \end{aligned}$$

(25)

where

$$\begin{aligned} \varTheta _3=\gamma _{40}^2\big (1+C_{<k_c>_N}\big ) + \big (\gamma _{42}^2+\gamma _{42}^3\big )\big (C_{<k_c>_N}^c+C_{<2k_c>_N}\big ) \end{aligned}$$

(26)

and

$$\begin{aligned} \varTheta _5=\gamma _{40}^4\big (C_{<k_c>_N} + 1\big ) +\gamma _{3}\gamma _2^c\big (C_{<3k_c>_N} + C_{<2k_c>_N}^c\big ). \end{aligned}$$

(27)

Note that this equation is also obtained from the fact that the vectors $W_\pm \,1$ and $W_5$ are linearly independent.

Hence, by combining equations (13) and (25), one obtains quintic S–L equation (16).