Pattern formation in a space- and time-discrete predator–prey system with a strong Allee effect

Abstract

The spatiotemporal dynamics of a space- and time-discrete predator–prey system is considered theoretically using both analytical methods and computer simulations. The prey is assumed to be affected by the strong Allee effect. We reveal a rich variety of pattern formation scenarios. In particular, we show that, in a predator–prey system with the strong Allee effect for prey, the role of space is crucial for species survival. Pattern formation is observed both inside and outside of the Turing domain. For parameters when the local kinetics is oscillatory, the system typically evolves to spatiotemporal chaos. We also consider the effect of different initial conditions and show that the system exhibits a spatiotemporal multistability. In a certain parameter range, the system dynamics is not self-organized but remembers the details of the initial conditions, which evokes the concept of long-living ecological transients. Finally, we show that our findings have important implications for the understanding of population dynamics on a fragmented habitat.

Appendix: A simple lattice system exhibiting unbounded predator growth

In order to demonstrate that the unbounded growth observed for μ _P  = 0 (see Fig. 8) is not a numerical artifact but is indeed a property of a predator–prey system on a lattice, we consider the following simple spatial model where the lattice is reduced to just three sites; see Fig. 16.

Fig. 16

A sketch of the simple lattice system. Predator is present at site c. Arrows show prey dispersal

For our purposes here, it is sufficient to identify at least one case when this simple system exhibit an unbounded predator growth. A more general case will be considered elsewhere. We therefore restrict our analysis to the following initial conditions:

$$ N_{a,0} = N_{b,0} = N_{c,0} = N_{2}^{*}, $$

(39)

$$ P_{a,0} = P_{b,0}=0, \quad P_{c,0} = P_0>0~. $$

(40)

For any t > 0, predator can only dwell in site c but, since it cannot disperse, not in sites a and b. Prey can be present in all three sites.

Also, for convenience, instead the original prey growth function f(N) = N ²/(1 + bN ²), we consider its piecewise-constant approximation (see Fig. 17):

$$ \tilde{f}(N) = N_{2}^{*}~H\left(N-N_{1}^{*}\right), $$

(41)

where H(z) is the Heaviside step function:

$$ H(z) = 0~~\mbox{for}~~z<0, \quad H(z) = 1~~\mbox{for}~~z\ge 0. $$

(42)

The property of the prey dynamics with the growth function $\tilde{f}$ is that, after one step in time, a population with any size $N<N_{1}^{*}$ is brought to extinction while a population with any size $N\ge N_{1}^{*}$ is brought to its carrying capacity $N_{2}^{*}$.

Fig. 17

Qualitative approximation of the prey growth function (solid curve) by a step function (blue dashed-and-dotted broken line), $N_{1}^{*}$ and $N_{2}^{*}$ are prey equilibria in the absence of predator

We point out that the step function is not a good quantitative approximation of the original growth function but it properly takes into account its qualitative features such as the position of the equilibria and the flatness of the curve in vicinity of $N_{2}^{*}$.

Since

$$ P_{c,t+1} = N_{c,t}^{^{\prime}}P_{c,t}~, $$

(43)

where $N_{c,t}^{^{\prime}}$ is the prey size at site c after dispersal, an unbounded growth occurs if $N_{c,t}^{^{\prime}}>1$ for any t > 0 (and $N_{c,t}^{^{\prime}}$ does not approach unity in the course of time).

It is readily seen that the dispersal stage will have no effect on the initial conditions. After the reaction stage, we obtain

$$ N_{a,1} = N_{b,1} = N_{2}^{*}, \quad N_{c,1} = N_{2}^{*}\exp(-P_0), $$

(44)

$$ P_{a,1} = P_{b,1}=0, \quad P_{c,1} = N_{2}^{*} P_0~. $$

(45)

The dispersal stage of the second step will not change the values of P but results in the following values of N:

$$ N_{a,1}^{^{\prime}} = \left(1 -\frac{\mu_N}{2}\right)N_{2}^{*} +\frac{\mu_N}{2} N_{2}^{*}\exp(-P_0), $$

(46)

$$ N_{b,1}^{^{\prime}} = \left(1 -\frac{\mu_N}{2}\right)N_{2}^{*} +\frac{\mu_N}{2} N_{2}^{*}\exp(-P_0), $$

(47)

$$ N_{c,1}^{^{\prime}} = \left(1-\mu_N\right) N_{2}^{*}\exp(-P_0) +\mu_N N_{2}^{*}. $$

(48)

Now, if we require

$$\begin{array}{rll} \mu_N N_{2}^{*} &<& 2\left(N_{2}^{*}\!-\! N_{1}^{*}\right) ~~\text{and} \nonumber\\ \mu_N N_{2}^{*} &>& \max\{1,N_{1}^{*}\}, \end{array}$$

(49)

then it is readily seen that

$$ \mbox{(a)}~N_{a,1}^{^{\prime}}=N_{b,1}^{^{\prime}} > N_{1}^{*} \quad \mbox{and} \quad \mbox{(b)}~N_{c,1}^{^{\prime}} > 1. $$

(50)

From Eqs. 17 and 50a, we obtain $N_{a,2} = N_{b,2} = N_{2}^{*}$. Condition 50b ensures predator growth, so that P _c,2 > P _c,1.

It is readily seen that $N_{a,t} = N_{b,t} = N_{2}^{*}$ for any t. For the dispersal stage of step (t + 1), taking into account Eq. 49, we then obtain:

$$ N_{a,t}^{^{\prime}} = N_{b,t}^{^{\prime}}~=~\left(1 -\frac{\mu_N}{2}\right)N_{2}^{*} +\frac{\mu_N}{2} N_{c,t}~>~N_{1}^{*}, $$

(51)

$$\begin{array}{rll} N_{c,t}^{^{\prime}} &=& \left(1-\mu_N\right) N_{c,t}+\mu_N N_{2}^{*} \\ &>&\mu_N N_{2}^{*}>\max\{1,N_{1}^{*}\}\ge1. \end{array}$$

(52)

Therefore, $N_{c,t}^{^{\prime}}>1$ for any t (and is separable from 1), which results in an unbounded predator growth at site c.