Time correlation function in systems with two coexisting biological species

We study a stochastic lattice model describing the dynamics of coexistence of two interacting biological species. The model comprehends the local processes of birth, death, and diffusion of individuals of each species and is grounded on interaction of the predator-prey type. The species coexistence can be of two types: With self-sustained coupled time oscillations of population densities and without oscillations. We perform numerical simulations of the model on a square lattice and analyze the temporal behavior of each species by computing the time correlation functions as well as the spectral densities. This analysis provides an appropriate characterization of the different types of coexistence. It is also used to examine linked population cycles in nature and in experiment.

Figure 1

Transitions of the predator-prey model. The three states are prey (1), predator (2), and empty (0). The allowed transitions obey the cyclic order shown.

Figure 2

(Color online) Average densities of prey (lower points) and predator (upper points) for $p=0.3$ and $c=0.05$ as a function of $D$.

Figure 3

(Color online) (a) Density of prey $x$ and density of predator $y$ versus $c$ for $p=0.3$ and $D=0.9$. (b) The quantity ${\left(1-x\right)}^{1/\beta }$ versus $c$, where $\beta =0.58$. The critical point occurs at $c_c=0.320\left(5\right)$.

Figure 4

(a) Value of the cross-correlation $r$ at the first maximum as a function of the parameter $c$, for $p=0.3$ and $D=0.9$. (b) Plot of $r^2$ as functions of $c$. The linear regression gives the value $c^{\ast }=0.278\left(5\right)$ for the transition from oscillatory to ordinary coexistence.

Figure 5

(Color online) (a) Densities of prey (lower curve) and predator (upper curve) as a function of time for $p=0.3$, $c=0.03$, and $D=0.9$. (b) Cross-correlation function between prey and predator. (c) Time autocorrelation function for prey. (d) Power spectral density for prey. The largest peak occurs at $\omega =0.0057$ which gives a period $T=1.10\times {10}^3$. The second and the third peaks occur at $\omega =0.0112$ and $\omega =0.016$. The slope of the tail equals $\alpha =2.0$.

Figure 6

(Color online) (a) Densities of prey (upper curve) and predator (lower curve) as a function of time for $p=0.3$, $c=0.15$, and $D=0.9$. (b) Cross-correlation function between prey and predator. (b) Time autocorrelation function for prey. (d) Power spectral density for prey. The largest peak occurs at $\omega =0.0102$ which gives a period $T=616$. The slope of the tail equals $\alpha =2.0$.

Figure 7

(Color online) (a) Densities of prey (upper curve) and predator (lower curve) as a function of time for $p=0.3$, $c=0.25$, and $D=0.9$. (b) Cross-correlation function between prey and predator. (c) Time autocorrelation function for prey. (d) Power spectral density for prey. The largest peak occurs at $\omega =0.007$ which gives a period $T=9\times {10}^2$. The slope of the tail equals $\alpha =2.0$.

Figure 8

(Color online) (a) Densities of prey (upper curve) and predator (lower curve) as a function of time for $p=0.3$, $c=0.29$, and $D=0.9$. (b) Cross-correlation function between prey and predator. (c) Time autocorrelation function for prey. (d) Power spectral density for prey. The slope of the tail equals $\alpha =2.0$.

Figure 9

(Color online) Simulational data obtained at $p=0.3$, $c=0.05$, and $D=0.9$. (a) Densities of prey as a function of time for $L=40$, 100, and 320, from bigger to smaller amplitudes. (b) Amplitude of prey density oscillations as a function of $\sqrt{N}$, where $N=L\times L$ is the total number of sites. The linear behavior of the log-log plot shows that $A\sim 1/\sqrt{N}$. (c) Time autocorrelation function for prey for $L=40$, 100, and 320. (d) Power spectral density for prey for $L=40$, 100, and 320.

Figure 10

(Color online) Predator density, $y$, versus prey density, $x$, corresponding to $p=0.3$ and $D=0.9$, for $c=0.03$, $c=0.15$, $c=0.25$, and $c=0.29$, from the left-hand side to the right-hand side, respectively.

Figure 11

(Color online) Oscillatory time behavior in the abundance of the snowshoe hare and the Canadian lynx in Canada. (a) Number of lynx ($y$, dotted curve) and number of hares ($x$, continuous curve) in thousands as a function of time in years (redrawn from Ref. [6]). (b) Cross-correlation function between lynx and hare. (c) Time autocorrelation function for hare. (d) Power spectral density for hare. (e) Time autocorrelation function for lynx. (f) Power spectral density for lynx.

Figure 12

(Color online) Oscillation in the populations of the host-parasite system azuki weevil and its parasite larval wasp [46]. (a) Population versus time in generations: Parasite population (dotted curve) and host population (continuous curve) (redrawn from Ref. [46]). (b) Cross-correlation function between host and parasite. (c) Autocorrelation for host. (d) Power spectral density of host. (e) Autocorrelation for parasite. (f) Power spectral density of parasite.