Increased persistence via asynchrony in oscillating ecological populations with long-range interaction

Understanding the influence of the structure of a dispersal network on the species persistence and modeling a realistic species dispersal in nature are two central issues in spatial ecology. A realistic dispersal structure which favors the persistence of interacting ecological systems was studied [M. D. Holland and A. Hastings, Nature (London) 456, 792 (2008)], where it was shown that a randomization of the structure of a dispersal network in a metapopulation model of prey and predator increases the species persistence via clustering, prolonged transient dynamics, and amplitudes of population fluctuations. In this paper, by contrast, we show that a deterministic network topology in a metapopulation can also favor asynchrony and prolonged transient dynamics if species dispersal obeys a long-range interaction governed by a distance-dependent power law. To explore the effects of power-law coupling, we take a realistic ecological model, namely, the Rosenzweig-MacArthur model in each patch (node) of the network of oscillators, and show that the coupled system is driven from synchrony to asynchrony with an increase in the power-law exponent. Moreover, to understand the relationship between species persistence and variations in power-law exponent, we compute a correlation coefficient to characterize cluster formation, a synchrony order parameter, and median predator amplitude. We further show that smaller metapopulations with fewer patches are more vulnerable to extinction as compared to larger metapopulations with a higher number of patches. We believe that the present work improves our understanding of the interconnection between the random network and the deterministic network in theoretical ecology.

Figure 1

All patches are connected to all patches in the network, and dispersal is bidirectional following the long-range interaction as modeled in Eq. (1). The color of the edges represents the power-law strength ($r^{-s}$) between patches with (a) $s=0.5$ and (b) $s=1.5$. Higher $s$ suggests that the species is less likely to move to further patches. Spatially separated patches are marked with solid circles. Here we consider an ecological network with 11 patches.

Figure 2

Time series of (a), (d), (g), (j) total predator amplitude and (b), (e), (h), (k) predator species, and (c), (f), (i), (l) phase portrait of predator vs prey for different cluster solutions. Local dynamics are governed by strong prey self-regulation $\theta =0.3$ and predator mortality rate $\eta =1$ comparable to prey birth rates. Dispersal rates are $d_h=2^{-5}$ and $d_p=2^{-6}$. Other parameters are (a)–(c) $s=0, \varphi =2, k=1$; (d)–(f) $s=0, \varphi =3, k=2$; (g)–(i) $s=1, \varphi =3, k=6$; and (j)–(l) $s=2, \varphi =3, k=11$.

Figure 3

Distribution of cluster states for different values of $s$: (a) $s=0$, (b) $s=0.3$, (c) $s=0.6$, (d) $s=0.9$, and (e) $s=1.2$. The local dynamics are governed by weak predation ($\varphi =2.7$), strong prey self-regulation ($\theta =0.3$), and predator mortality rate ($\eta =1$) comparable to prey birth rates. The spatial parameters are $d_h=2^{-5}$ and $d_p=2^{-6}$. The number of clusters increases with increasing values of $s$. The above results in each panel correspond to 100 numerical simulations carried out independently.

Figure 4

Synchrony order parameter ($\sigma$) with variations in $s$. The other parameters are $d_h=2^{-10}, d_p=2^{-10}, \varphi =2, \eta =1$, and $\theta =0.3$. The network size is 11. Points are fitted by a polynomial of degree 4 (blue curve). The synchrony order parameter ($\sigma$) decreases with increasing $s$. The above result corresponds to 100 numerical simulations carried out independently.

Figure 5

Predator amplitude and transient duration: (a), (c), (e) median total predator amplitude during transient (triangles, solid lines) and asymptotic (open circles, dashed lines) solution phases for a regular network; (b), (d), (f) mean transient fraction for different parameters. The cyan line in (a), (c), and (e) corresponds to a global synchronous solution for the chosen parameter values: (a), (b) $\varphi =3.0, \eta =1, \theta =0.3, d_h=2^{-5}, d_p=2^{-6}$; (c), (d) $\eta =1, \theta =0.3, d_h=2^{-5}, d_p=2^{-6}$; and (e), (f) $\varphi =3.0, \eta =1, \theta =0.3, d_h=2^{-5}$. The color coding is as follows: $s=0.0$ (dark blue), $s=0.5$ (green), $s=1.0$ (red), $s=1.5$ (light blue), and $s=2.0$ (pink). The above result corresponds to 150 numerical simulations carried out independently.

Figure 6

Variation of cluster solutions with spatial and dispersal parameters: (a) dispersal rates are $d_h=2^{-5}$ and $d_p=2^{-6}$ and (b) prey dispersal rate is $d_h=2^{-5}$. Local dynamics are governed by weak predation $\varphi =3.0$, strong prey self-regulation $\theta =0.3$, and predator mortality rate $\eta =1$. Each grid summarizes 120 numerical simulations carried out independently.

Figure 7

Predator amplitude and transient time duration as a function of network size. Median total predator amplitude during transient (triangles, solid lines) and asymptotic (open circles, dashed lines) solution phases for a regular network. The cyan line corresponds to a global synchronous solution for the chosen parameter values: $\eta =1, \theta =0.3, d_h=2^{-5}$, and $d_p=2^{-6}$. (a), (b) $\varphi =3, s=0$; (c), (d) $\varphi =3, s=0.3$; (e), (f) $\varphi =3, s=0.5$; and (g), (h) $\varphi =3, s=1.0$. The above result corresponds to 150 numerical simulations carried out independently.

Figure 8

Distribution of cluster states for different values of $s$: (a) $s=0$, (b) $s=0.3$, (c) $s=0.6$, (d) $s=0.9$, and (e) $s=1.2$. The local dynamics are governed by prey growth rate $(\alpha =1/3)$, predation rate $(\beta =4/3)$, predator efficiency $(\gamma =3/4)$, and predator mortality rate $(\delta =1)$. The spatial parameters are $d_h=2^{-5}$ and $d_p=2^{-6}$. The number of clusters increases with increasing values of $s$. The above results in each panel correspond to 100 numerical simulations carried out independently.

Figure 9

Predator amplitude and transient phase duration: (a) median total predator amplitude during transient (triangles, solid lines) and asymptotic (open circles, dashed lines) solution phases for a regular network and (b) mean transient fraction for different parameters. The chosen parameter values are $\alpha =1/3, \beta =4/3, \gamma =3/4, \delta =1, d_h=2^{-8}$, and $d_p=2^{-7}$. The above result corresponds to 150 numerical simulations carried out independently.