Fig. 1. (a) An example of a travelling wave family predicted by a predator–prey reaction–diffusion model (Eqs. (1a) and (1b) with reaction kinetics (2) and density-dependent dispersal function (4)). The parameter values are σ=0.15, μ=0.05, and κ=0.2, D_u,max=10^0.5, D_u,min=10^−0.5 (implying D_v=1) and m=−100 (these parameter values are defined in the text). (b) and (c) show periodic travelling waves arising from two different selection mechanisms in simulations of the same model as in (a) and, hence, they are selected from the same wave family; marked with labelled crosses in (a). A landscape obstacle is assumed in (b), with (u, v)=(0, 0) at the left boundary (simulating an inhospitable habitat at x<0) and du/dx=dv/dx=0 at the right boundary. This simulation started with random initial predator and prey densities. (c) Simulates predators invading a prey population, with du/dx=dv/dx=0 assumed at both boundaries. This simulation started with the prey-only steady state, (u, v)=(1, 0), throughout the domain except the left boundary, which started with (u, v)=(1, 1). Note that the invasion front (where prey density sharply declines from (u=1)) has travelled to the right of the domain in this scenario. Animations of the dynamics in this figure, and other figures in this paper, can be generated and explored using the custom made software tool that is downloadable from http://research.microsoft.com/ero/biosciences/software.aspx.

Fig. 2. The shapes of the relationship between population density, u, and the diffusion coefficient, D_u(u) (Eq. (4)), for (a) the lambda–omega and (b) predator–prey reaction kinetics (Eqs. (3a) and (2a), respectively) and for different values of m, with D_u(u_s)=1, D_u,max=10^(0.5), and D_u,min=10^(−0.5). Note that D_u varies on a log₁₀ scale. Positive and negative values of m correspond to positive and negative density-dependent dispersal, respectively.

Fig. 3. Comparison of varying the gradient of density-dependent dispersal m, between −100 (strong negative density-dependence) and 100 (strong positive density-dependence) with varying the ratio of constant diffusion coefficients α=D(u_s)/D_v (m=0) between 0.01 and 100 (log scale), on the shape of the travelling wave family predicted by Eqs. (1a) and (1b). (a) and (b) have lambda–omega kinetics (3) with ω₀=1.5 and ω₁=0.5. (c) and (d) have predator–prey kinetics (2) with σ=0.15, μ=0.05, and κ=0.2. α=1 in (a) and (c). m=0 in (b) and (d). Dispersal parameter values are D_u,max=10^0.5 and D_u,min=10^−0.5, implying that D_v=1. To aid interpretation we have rescaled both wavelength and time period, which simply relabelled the axes. We divided the time period by the time period predicted by the non-spatial models (corresponding to infinite dispersal rates in Eqs. (1a) and (1b)), and we divided the wavelength by that at the origin of the wave family when m=0 and α=1.

Fig. 4. Comparison of varying the gradient of density-dependent dispersal m, between −100 (strong negative density-dependence) and 100 (strong positive density-dependence) with varying the ratio of constant diffusion coefficients α=D_u/D_v (m=0) between 0.01 and 100, on the wavelengths of periodic travelling waves (symbols) picked out by simulations of Eqs. (1a) and (1b). Note that α determines D_u and D_v, because our non-dimensionalization implies that D_uD_v=1. (a) and (b) have lambda–omega kinetics (3) and (c) and (d) have predator–prey kinetics (2). α=1 in (a) and (c). m=0 in (b) and (d). Parameter values are the same as those detailed in the legend to Fig. 3. Filled circles denote the wavelengths of periodic travelling waves resulting from simulations with a landscape obstacle. In all of these cases the waves travel away from the obstacle edge (as demonstrated in Fig. 1(b)). Triangles denote the wavelengths of periodic travelling waves resulting from predator invasion into a prey population. Upwards pointing triangles denote waves moving to the left, in the opposite direction to the invasion front, and downwards pointing triangles denote waves moving to the right. Superimposed on the graphs are contour lines of fixed time period (thin lines), and the minimum wavelength (thick lines) from the analysis of the travelling wave families. To aid interpretation we have rescaled both wavelength and time period, which simply relabelled the axes, as detailed in the legend to Fig. 3.

Fig. C.1. Comparison of the effects of varying the gradient of density-dependent dispersal m, between −100 (strong negative density-dependence) and 100 (strong positive density-dependence) with varying the ratio of constant diffusion coefficients α=D_u/D_v (m=0) between 0.01 and 100, on the wavelengths of periodic travelling waves (symbols) picked out by simulations of Eqs. (1a) and (1b). Note that α determines D_u and D_v, because our non-dimensionalization implies that D_uD_v=1. The figure is exactly as in Fig. 4 of the main text except that here the results of the stability analysis are included, with stable waves lying within the grey shaded region and unstable waves lying within the white region above the thick black line. This line is the boundary of the region in which periodic travelling waves exist. Note that for the predator–prey equations some selected waves lie in the unstable region. In these cases, spatiotemporal irregularities develop behind a large region of waves, see for example Fig. C.2.

Fig. C.2. Unstable periodic travelling waves arising from a landscape obstacle (at x=0) in a simulation of Eqs. (1a) and (1b) with predator–prey reaction kinetics (2). The boundary conditions are (u, v)=(0, 0) at the left boundary (simulating the edge of an obstacle, or of an inhospitable habitat) and du/dx=dv/dx=0 at the right boundary. This simulation started (at t=0) with predator and prey densities chosen randomly from a uniform distribution between 0 and 1. Parameter values are σ=0.15, μ=0.05, and κ=0.2, D_u,max=10^0.5 and D_u,min=10^−0.5 (implying D_v=1). We also assume strong positive density-dependent dispersal, m=100, corresponding to the right-most circle in Fig. C.1(c).