Fig 1.
The grey shaded region represents the region of stability for the equilibrium of the generalized Lotka-Volterra model (3).
The stability criterion (10) is plotted in terms of the log sensitivities of the attack rate f(h, p) to the prey (fh) and predator (fp) densities. The yellow line within the stability region separates the regions of negative real eigenvalues of the Jacobian matrix and complex eigenvalues with negative real parts. For this plot, the prey’s growth rate is assumed to be r = 2 per unit time and γ = 1 per unit time.
Fig 2.
Noise in the fluctuations of population densities as determined by (25) plotted for increasing fh for fp = 0.
Realizations of the prey’s growth rate, population densities of the prey and the predator as obtained by simulating (11) and (12) using are shown for fh = 0.5 (left) and fh = 10 (right). Each realization is normalized to its initial value so that it starts at a value of one. Note that for large values of fh fluctuations in the predator density remain pronounced even though fluctuations in the prey density are minimal. For this plot,
,
,
.
Fig 3.
Noise in the fluctuations of prey density as determined by (23) plotted as a function of fp.
Noise is minimized at an intermediate value of fp and stochastic realizations of the prey and predator densities are shown for three different values of fp. For this plot, , fh = 0.5, σ = 1 and
. Note from Fig 1 that as fp increases from −1 to 0.75 (for a fixed value of fh = 0.5), one goes from real to complex eigenvalues that is reflected in the pronounced oscillatory dynamics of population densities as seen in the simulations on the right.
Fig 4.
Pearson’s correlation coefficient between the predator and the prey population densities as predicted by (31) for varying levels of fp with fp < 0 (fp > 0) driving a positive (negative) correlation.
Sample trajectory paths are shown for fp = 1 and fp = −1. Other parameters taken as , fh = 0.6 and
.