Time-dependent localized patterns in a predator-prey model

TL;DR

A novel mechanism whereby disconnected segments of oscillatory states zip up into a continuous snaking branch of time-periodic localized states, some of which are stable, is uncovered using two-parameter continuation.

Abstract

Numerical continuation is used to compute solution branches in a two-component reaction-diffusion model of Leslie--Gower type. %in the vicinity of a Turing-Hopf interaction. Two regimes are studied in detail. In the first, the homogeneous state loses stability to supercritical spatially uniform oscillations, followed by a subcritical steady state bifurcation of Turing type. The latter leads to spatially localized states embedded in an oscillating background that bifurcate from snaking branches of localized steady states. Using two-parameter continuation we uncover a novel mechanism whereby disconnected segments of oscillatory states zip up into a continuous snaking branch of time-periodic localized states, some of which are stable. In the second, the homogeneous state loses stability to supercritical Turing patterns, but steady spatially localized states embedded either in the homogeneous state or in a small amplitude Turing state are nevertheless present. We show that such behavior is possible when sideband Turing states are strongly subcritical and explain why this is so in the present model. In both cases the observed behavior differs significantly from that expected on the basis of a supercritical primary bifurcation.

Figures (8)

• Figure 1: (a) Hopf and Turing instability curves for the homogeneous solution $s_4$ of \ref{['eq1']} in the $(b,\delta)$ plane when $(a,\alpha,\beta,D)=(1,2,0.5,0.029)$ together with the existence regions for localized patterns. (b) Spectra $\lambda(k)$ at the locations indicated in (a). (c) Bifurcation curves for $D=0.07$ (top) and $D=0.01$ (bottom) showing the dependence of the Turing instability (black curves) on the parameter $D$. The Hopf bifurcation (red curve) is independent of $D$.
• Figure 2: (a) Bifurcation diagram for $\ell=50$ with $(\delta,\alpha,\beta,a,D)=(0.07,2,0.5,1,0.029)$ showing $\|u\|_2$ defined in (\ref{['n1']}) and (\ref{['n2']}) as a function the parameter $b$ starting from $b=0.78$. (b) Sample time-periodic solutions ($u$ component only) shown in a space-time plot over one temporal period corresponding to the locations labeled in (a). (c) Sample steady solutions ($u$ component only). In (a) Hopf bifurcation points (HPs) are marked by $\diamond$, steady state BPs by $\circ$, and fold points (FPs) by $\times$. Not all HPs, BPs and FPs are shown. For steady states, thick (thin) lines indicate stable (unstable) solution branches. The trivial state $s_4$ corresponds to $u=0$. Hopf branches H1 and H2 from this state are shown in brown and yellow, with sample solutions A and B. The primary Turing branch T1 is shown in blue; the 6th Turing branch (T6, dark blue) reaches farthest to the left. A snaking branch of LS (S1, orange) bifurcates from the first BP on T1 and reconnects to T6, sample solutions in (c). Additional HPs on S1 lead to branches with sample solutions (C-F).
• Figure 3: Zipping up the Turing-Hopf segments from Fig. \ref{['f1']} into a Turing-Hopf snake on increasing $D$. (a) HPC of HP2 (brown), HP3 (dark green) and HP4 (light green) from Fig. \ref{['f1']} showing the HP location in the $(D,b)$ plane, together with the corresponding Hopf frequency $\omega$. (b) HPC of HP2 (brown), and FPC of FP2 and FP3 (orange) together with sample solutions I-III used to initialize continuation in $b$ in panels (c) and (d). (c) Bifurcation diagram for $D=0.02965$ corresponding to point I in panel (b); the inset shows the LS branch (orange) near the first fold, shortly before HP2 and HP3 annihilate (at $D=D_1\approx 0.02968$) together with the disconnected MM branches (brown and green). The red branch (inset) shows the MM branches at $D=0.0297$, i.e., after reconnection. As a result the large $b$ MM branch now connects to the point HP4 on S1. To generate the red branch we take point A from the green branch at $D=0.02965$, set $D=0.0297$, and continue in $b$. (d) Bifurcation diagram for $D\approx 0.03$ corresponding to point II in panel (b). Stability on S1 is indicated by thick lines; stability on the other branches is not indicated (see text).
• Figure 4: As for Figs. \ref{['f3b']}(c,d) but for $D=0.033$, corresponding to point III in Fig. \ref{['f3b']}(b): the whole Turing-Hopf snake has zipped up, generating a snaking branch of time-periodic states terminating near the top of S1 (solutions A-C). Stability is not indicated. The only other HP left on S1 is the one at the bottom leading to the (red, unstable) MM branch with a nonuniform background oscillation (solution D here and solution C in Fig. \ref{['f3b']}(d)). The figure also shows other branches at $D=0.033$, namely H1 (brown), H2 (yellow), H3 (violet, with sample solution E), as well as T1 (blue) and T4 (4th Turing branch from $s_4$, dark blue with sample solution F), the termination of S1 for this value of $D$.
• Figure 5: DNS from marked points (+small perturbation) in Fig. \ref{['f3b2']}(a). (a) Starting from C with fast convergence to H1. (b) Starting from B with slow convergence to H1. (c) Starting from B but setting $b=b+0.015$ (right of the TH snake), yielding (stick--slip) convergence to T4. (d) Starting from B but setting $b=b+0.007$, yielding convergence to a stable MM in the TH snake.

Theorems & Definitions (2)

• Remark 2.1
• Remark 3.1