chapter  14
6 Pages

Noise-induced pattern transitions

To begin with, the influence of parametric noise on the pattern formation in the Rosenzweig-MacArthur (1963) prey-predator model is investigated. The latter model has been used by Scheffer (1991a) for specifying the phytoplankton-zooplankton dynamics in a shallow lake under the control of nutrient density and of planktivorous fish stock. Structures in a deterministic environment have been presented by Malchow (1993, 2000a). The model reads in dimensionless quantities


∂t = r u (1− u)− au

1 + bu v + du Δu , (14.1)


∂t =


1 + bu v −mv v − g

1 + h2v2 f + dv Δv . (14.2)

The environmental heterogeneity is described through a simple approach: the considered L × 2L model area is divided into three habitats of sizes L × L/2, L × L and L × L/2 respectively; cf. Figure 14.1. The following model parameters have been chosen for the simulations, cf. Pascual (1993); Malchow et al. (2000, 2002):

〈 r 〉 = 1 , a = b = 5 , g = h = 10 , m = 0.6 , f = 0 , (14.3)

L = 100 , x ∈ [0, L] , y ∈ [0, 2L] , du = dv = 5× 10−2 . (14.4)

〈 r 〉 is the spatially averaged prey growth rate r(x, y) whereas m stands for the spatiotemporal mean of the noisy predator mortality rate mv(x, y, t). The value of mv is randomly chosen at each space point and each unit time step from a truncated normal distribution between I = 0 and 15% of m, i.e. m(x, y, t) = m [ 1+ I−rndm (2I) ] with rndm (z) as a random number between 0 and z. The upper layer has double mean phytoplankton productivity 2〈 r 〉; the bottom layer only 60% of 〈 r 〉. Both are coupled by the middle habitat with linearly decreasing productivity. The latter gradient reflects assumptions by Pascual (1993). The chosen model parameters generate limit cycles at each space point, i.e., one has a kind of continuous chain of diffusively

coupled nonlinear oscillators. The spatial setting yields a fast spatially uniform prey-predator limit cycle in the top habitat, continuously changing into quasiperiodic and chaotic oscillations and waves along the productivity gradient in the middle, coupled to slow spatially uniform limit cycle oscillations in the lower layer.