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Abstract. We study how movement between habitat patches changes the long-run behaviour of a two-species predator–prey system. In a single patch the populations cycle with growing amplitude and eventually collapse. Letting a small fraction of each population disperse between patches damps those cycles and lets both species persist. We derive a simple condition on the dispersal rate that separates persistence from collapse, and test it against simulations of up to one hundred coupled patches.
Ecologists have long suspected that spatial structure stabilises interacting populations [1, 2]. Classic mite experiments showed that predators and prey coexisted far longer when the habitat was divided into patches than when it was a single well-mixed arena [3]. Theory has been slower to explain why.
Most models assume that dispersal is either negligible or instantaneous. Neither describes real populations, where individuals move at finite rates and arrive at neighbouring patches already out of phase with the residents [4]. Here we treat dispersal as a continuous flow and ask how large that flow must be before the benefit appears.
2 ModelEach patch holds a prey density and a predator density. Within a patch, prey grow logistically and are consumed at a rate proportional to the product of the two densities; predators convert consumed prey into offspring and die at a constant rate [5].
3 ResultsFor two identical patches the coexistence state becomes stable once the dispersal rate exceeds a critical value that grows with the square of the oscillation frequency [6]. Below it, the patches oscillate in step and collapse together; above it, they settle into out-of-phase cycles that never reach zero.
In networks of up to one hundred patches the same threshold holds to within ten per cent, although persistence times are longer when each patch has more neighbours.
4 DiscussionDispersal is most valuable at intermediate rates: too little leaves each patch to its own fate, and too much merges them into one large, unstable patch.
We study how movement between habitat patches changes the long-run behaviour of a two-species predator–prey system. In a single patch the populations cycle with growing amplitude and eventually collapse. Letting a small fraction of each population disperse between patches damps those cycles and lets both species persist. We derive a simple condition on the dispersal rate that separates persistence from collapse, and test it against simulations of up to one hundred coupled patches.
1 IntroductionEcologists have long suspected that spatial structure stabilises interacting populations [1, 2]. Classic mite experiments showed that predators and prey coexisted far longer when the habitat was divided into patches than when it was a single well-mixed arena [3]. Theory has been slower to explain why.
Most models assume that dispersal is either negligible or instantaneous. Neither describes real populations, where individuals move at finite rates and arrive at neighbouring patches already out of phase with the residents [4]. Here we treat dispersal as a continuous flow and ask how large that flow must be before the benefit appears.
2 ModelEach patch holds a prey density and a predator density. Within a patch, prey grow logistically and are consumed at a rate proportional to the product of the two densities; predators convert consumed prey into offspring and die at a constant rate [5].
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For two identical patches the coexistence state becomes stable once the dispersal rate exceeds a critical value that grows with the square of the oscillation frequency [6]. Below it, the patches oscillate in step and collapse together; above it, they settle into out-of-phase cycles that never reach zero.
Figure 2. Prey and predator density, with and without dispersal.Listening
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We study how movement between habitat patches changes the long-run behaviour of a two-species predator–prey system. In a single patch the populations cycle with growing amplitude and eventually collapse. Letting a small fraction of each population disperse between patches damps those cycles and lets both species persist. We derive a simple condition on the dispersal rate that separates persistence from collapse, and test it against simulations of up to one hundred coupled patches.
1 IntroductionEcologists have long suspected that spatial structure stabilises interacting populations [1, 2]. Classic mite experiments showed that predators and prey coexisted far longer when the habitat was divided into patches.
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We study how movement between habitat patches changes the long-run behaviour of a two-species predator–prey system. In a single patch the populations cycle with growing amplitude and eventually collapse. Letting a small fraction of each population disperse between patches damps those cycles and lets both species persist. We derive a simple condition on the dispersal rate that separates persistence from collapse, and test it against simulations of up to one hundred coupled patches.
1 IntroductionEcologists have long suspected that spatial structure stabilises interacting populations [1, 2]. Classic mite experiments showed that predators and prey coexisted far longer when the habitat was divided into patches than when it was a single well-mixed arena [3]. Theory has been slower to explain why.
Most models assume that dispersal is either negligible or instantaneous. Neither describes real populations, where individuals move at finite rates and arrive at neighbouring patches already out of phase with the residents [4]. Here we treat dispersal as a continuous flow and ask how large that flow must be before the benefit appears.
2 ModelEach patch holds a prey density and a predator density. Within a patch, prey grow logistically and are consumed at a rate proportional to the product of the two densities; predators convert consumed prey into offspring and die at a constant rate [5].
We study how movement between habitat patches changes the long-run behaviour of a two-species predator–prey system. In a single patch the populations cycle with growing amplitude and eventually collapse. Letting a small fraction of each population disperse between patches damps those cycles and lets both species persist. We derive a simple condition on the dispersal rate that separates persistence from collapse, and test it against simulations of up to one hundred coupled patches.
1 IntroductionEcologists have long suspected that spatial structure stabilises interacting populations [1, 2]. Classic mite experiments showed that predators and prey coexisted far longer when the habitat was divided into patches than when it was a single well-mixed arena [3]. Theory has been slower to explain why.
Most models assume that dispersal is either negligible or instantaneous. Neither describes real populations, where individuals move at finite rates and arrive at neighbouring patches already out of phase with the residents [4]. Here we treat dispersal as a continuous flow and ask how large that flow must be before the benefit appears.
2 ModelEach patch holds a prey density and a predator density. Within a patch, prey grow logistically and are consumed at a rate proportional to the product of the two densities; predators convert consumed prey into offspring and die at a constant rate [5].
We study how movement between habitat patches changes the long-run behaviour of a two-species predator–prey system. In a single patch the populations cycle with growing amplitude and eventually collapse. Letting a small fraction of each population disperse between patches damps those cycles and lets both species persist. We derive a simple condition on the dispersal rate that separates persistence from collapse, and test it against simulations of up to one hundred coupled patches.
1 IntroductionEcologists have long suspected that spatial structure stabilises interacting populations [1, 2]. Classic mite experiments showed that predators and prey coexisted far longer when the habitat was divided into patches than when it was a single well-mixed arena [3]. Theory has been slower to explain why.
Most models assume that dispersal is either negligible or instantaneous. Neither describes real populations, where individuals move at finite rates and arrive at neighbouring patches already out of phase with the residents [4]. Here we treat dispersal as a continuous flow and ask how large that flow must be before the benefit appears.
2 ModelEach patch holds a prey density and a predator density. Within a patch, prey grow logistically and are consumed at a rate proportional to the product of the two densities; predators convert consumed prey into offspring and die at a constant rate [5].
We study how movement between habitat patches changes the long-run behaviour of a two-species predator–prey system. In a single patch the populations cycle with growing amplitude and eventually collapse. Letting a small fraction of each population disperse between patches damps those cycles and lets both species persist. We derive a simple condition on the dispersal rate that separates persistence from collapse, and test it against simulations of up to one hundred coupled patches.
1 IntroductionEcologists have long suspected that spatial structure stabilises interacting populations [1, 2]. Classic mite experiments showed that predators and prey coexisted far longer when the habitat was divided into patches than when it was a single well-mixed arena [3]. Theory has been slower to explain why.
Most models assume that dispersal is either negligible or instantaneous. Neither describes real populations, where individuals move at finite rates and arrive at neighbouring patches already out of phase with the residents [4]. Here we treat dispersal as a continuous flow and ask how large that flow must be before the benefit appears.
2 ModelEach patch holds a prey density and a predator density. Within a patch, prey grow logistically and are consumed at a rate proportional to the product of the two densities; predators convert consumed prey into offspring and die at a constant rate [5].
We study how movement between habitat patches changes the long-run behaviour of a two-species predator–prey system. In a single patch the populations cycle with growing amplitude and eventually collapse. Letting a small fraction of each population disperse between patches damps those cycles and lets both species persist. We derive a simple condition on the dispersal rate that separates persistence from collapse, and test it against simulations of up to one hundred coupled patches.
1 IntroductionEcologists have long suspected that spatial structure stabilises interacting populations [1, 2]. Classic mite experiments showed that predators and prey coexisted far longer when the habitat was divided into patches than when it was a single well-mixed arena [3]. Theory has been slower to explain why.
Most models assume that dispersal is either negligible or instantaneous. Neither describes real populations, where individuals move at finite rates and arrive at neighbouring patches already out of phase with the residents [4]. Here we treat dispersal as a continuous flow and ask how large that flow must be before the benefit appears.
2 ModelEach patch holds a prey density and a predator density. Within a patch, prey grow logistically and are consumed at a rate proportional to the product of the two densities; predators convert consumed prey into offspring and die at a constant rate [5].
We study how movement between habitat patches changes the long-run behaviour of a two-species predator–prey system. In a single patch the populations cycle with growing amplitude and eventually collapse. Letting a small fraction of each population disperse between patches damps those cycles and lets both species persist. We derive a simple condition on the dispersal rate that separates persistence from collapse, and test it against simulations of up to one hundred coupled patches.
1 IntroductionEcologists have long suspected that spatial structure stabilises interacting populations [1, 2]. Classic mite experiments showed that predators and prey coexisted far longer when the habitat was divided into patches than when it was a single well-mixed arena [3]. Theory has been slower to explain why.
Most models assume that dispersal is either negligible or instantaneous. Neither describes real populations, where individuals move at finite rates and arrive at neighbouring patches already out of phase with the residents [4]. Here we treat dispersal as a continuous flow and ask how large that flow must be before the benefit appears.
2 ModelEach patch holds a prey density and a predator density. Within a patch, prey grow logistically and are consumed at a rate proportional to the product of the two densities; predators convert consumed prey into offspring and die at a constant rate [5].
We study how movement between habitat patches changes the long-run behaviour of a two-species predator–prey system. In a single patch the populations cycle with growing amplitude and eventually collapse. Letting a small fraction of each population disperse between patches damps those cycles and lets both species persist. We derive a simple condition on the dispersal rate that separates persistence from collapse, and test it against simulations of up to one hundred coupled patches.
1 IntroductionEcologists have long suspected that spatial structure stabilises interacting populations [1, 2]. Classic mite experiments showed that predators and prey coexisted far longer when the habitat was divided into patches than when it was a single well-mixed arena [3]. Theory has been slower to explain why.
Most models assume that dispersal is either negligible or instantaneous. Neither describes real populations, where individuals move at finite rates and arrive at neighbouring patches already out of phase with the residents [4]. Here we treat dispersal as a continuous flow and ask how large that flow must be before the benefit appears.
2 ModelEach patch holds a prey density and a predator density. Within a patch, prey grow logistically and are consumed at a rate proportional to the product of the two densities; predators convert consumed prey into offspring and die at a constant rate [5].
We study how movement between habitat patches changes the long-run behaviour of a two-species predator–prey system. In a single patch the populations cycle with growing amplitude and eventually collapse. Letting a small fraction of each population disperse between patches damps those cycles and lets both species persist. We derive a simple condition on the dispersal rate that separates persistence from collapse, and test it against simulations of up to one hundred coupled patches.
1 IntroductionEcologists have long suspected that spatial structure stabilises interacting populations [1, 2]. Classic mite experiments showed that predators and prey coexisted far longer when the habitat was divided into patches than when it was a single well-mixed arena [3]. Theory has been slower to explain why.
Most models assume that dispersal is either negligible or instantaneous. Neither describes real populations, where individuals move at finite rates and arrive at neighbouring patches already out of phase with the residents [4]. Here we treat dispersal as a continuous flow and ask how large that flow must be before the benefit appears.
2 ModelEach patch holds a prey density and a predator density. Within a patch, prey grow logistically and are consumed at a rate proportional to the product of the two densities; predators convert consumed prey into offspring and die at a constant rate [5].
We study how movement between habitat patches changes the long-run behaviour of a two-species predator–prey system. In a single patch the populations cycle with growing amplitude and eventually collapse. Letting a small fraction of each population disperse between patches damps those cycles and lets both species persist. We derive a simple condition on the dispersal rate that separates persistence from collapse, and test it against simulations of up to one hundred coupled patches.
1 IntroductionEcologists have long suspected that spatial structure stabilises interacting populations [1, 2]. Classic mite experiments showed that predators and prey coexisted far longer when the habitat was divided into patches than when it was a single well-mixed arena [3]. Theory has been slower to explain why.
Most models assume that dispersal is either negligible or instantaneous. Neither describes real populations, where individuals move at finite rates and arrive at neighbouring patches already out of phase with the residents [4]. Here we treat dispersal as a continuous flow and ask how large that flow must be before the benefit appears.
2 ModelEach patch holds a prey density and a predator density. Within a patch, prey grow logistically and are consumed at a rate proportional to the product of the two densities; predators convert consumed prey into offspring and die at a constant rate [5].
We study how movement between habitat patches changes the long-run behaviour of a two-species predator–prey system. In a single patch the populations cycle with growing amplitude and eventually collapse. Letting a small fraction of each population disperse between patches damps those cycles and lets both species persist. We derive a simple condition on the dispersal rate that separates persistence from collapse, and test it against simulations of up to one hundred coupled patches.
1 IntroductionEcologists have long suspected that spatial structure stabilises interacting populations [1, 2]. Classic mite experiments showed that predators and prey coexisted far longer when the habitat was divided into patches than when it was a single well-mixed arena [3]. Theory has been slower to explain why.
Most models assume that dispersal is either negligible or instantaneous. Neither describes real populations, where individuals move at finite rates and arrive at neighbouring patches already out of phase with the residents [4]. Here we treat dispersal as a continuous flow and ask how large that flow must be before the benefit appears.
2 ModelEach patch holds a prey density and a predator density. Within a patch, prey grow logistically and are consumed at a rate proportional to the product of the two densities; predators convert consumed prey into offspring and die at a constant rate [5].
We study how movement between habitat patches changes the long-run behaviour of a two-species predator–prey system. In a single patch the populations cycle with growing amplitude and eventually collapse. Letting a small fraction of each population disperse between patches damps those cycles and lets both species persist. We derive a simple condition on the dispersal rate that separates persistence from collapse, and test it against simulations of up to one hundred coupled patches.
1 IntroductionEcologists have long suspected that spatial structure stabilises interacting populations [1, 2]. Classic mite experiments showed that predators and prey coexisted far longer when the habitat was divided into patches than when it was a single well-mixed arena [3]. Theory has been slower to explain why.
Most models assume that dispersal is either negligible or instantaneous. Neither describes real populations, where individuals move at finite rates and arrive at neighbouring patches already out of phase with the residents [4]. Here we treat dispersal as a continuous flow and ask how large that flow must be before the benefit appears.
2 ModelEach patch holds a prey density and a predator density. Within a patch, prey grow logistically and are consumed at a rate proportional to the product of the two densities; predators convert consumed prey into offspring and die at a constant rate [5].
We study how movement between habitat patches changes the long-run behaviour of a two-species predator–prey system. In a single patch the populations cycle with growing amplitude and eventually collapse. Letting a small fraction of each population disperse between patches damps those cycles and lets both species persist. We derive a simple condition on the dispersal rate that separates persistence from collapse, and test it against simulations of up to one hundred coupled patches.
1 IntroductionEcologists have long suspected that spatial structure stabilises interacting populations [1, 2]. Classic mite experiments showed that predators and prey coexisted far longer when the habitat was divided into patches than when it was a single well-mixed arena [3]. Theory has been slower to explain why.
Most models assume that dispersal is either negligible or instantaneous. Neither describes real populations, where individuals move at finite rates and arrive at neighbouring patches already out of phase with the residents [4]. Here we treat dispersal as a continuous flow and ask how large that flow must be before the benefit appears.
2 ModelEach patch holds a prey density and a predator density. Within a patch, prey grow logistically and are consumed at a rate proportional to the product of the two densities; predators convert consumed prey into offspring and die at a constant rate [5].
We study how movement between habitat patches changes the long-run behaviour of a two-species predator–prey system. In a single patch the populations cycle with growing amplitude and eventually collapse. Letting a small fraction of each population disperse between patches damps those cycles and lets both species persist. We derive a simple condition on the dispersal rate that separates persistence from collapse, and test it against simulations of up to one hundred coupled patches.
1 IntroductionEcologists have long suspected that spatial structure stabilises interacting populations [1, 2]. Classic mite experiments showed that predators and prey coexisted far longer when the habitat was divided into patches than when it was a single well-mixed arena [3]. Theory has been slower to explain why.
Most models assume that dispersal is either negligible or instantaneous. Neither describes real populations, where individuals move at finite rates and arrive at neighbouring patches already out of phase with the residents [4]. Here we treat dispersal as a continuous flow and ask how large that flow must be before the benefit appears.
2 ModelEach patch holds a prey density and a predator density. Within a patch, prey grow logistically and are consumed at a rate proportional to the product of the two densities; predators convert consumed prey into offspring and die at a constant rate [5].
We study how movement between habitat patches changes the long-run behaviour of a two-species predator–prey system. In a single patch the populations cycle with growing amplitude and eventually collapse. Letting a small fraction of each population disperse between patches damps those cycles and lets both species persist. We derive a simple condition on the dispersal rate that separates persistence from collapse, and test it against simulations of up to one hundred coupled patches.
1 IntroductionEcologists have long suspected that spatial structure stabilises interacting populations [1, 2]. Classic mite experiments showed that predators and prey coexisted far longer when the habitat was divided into patches than when it was a single well-mixed arena [3]. Theory has been slower to explain why.
Most models assume that dispersal is either negligible or instantaneous. Neither describes real populations, where individuals move at finite rates and arrive at neighbouring patches already out of phase with the residents [4]. Here we treat dispersal as a continuous flow and ask how large that flow must be before the benefit appears.
2 ModelEach patch holds a prey density and a predator density. Within a patch, prey grow logistically and are consumed at a rate proportional to the product of the two densities; predators convert consumed prey into offspring and die at a constant rate [5].
We study how movement between habitat patches changes the long-run behaviour of a two-species predator–prey system. In a single patch the populations cycle with growing amplitude and eventually collapse. Letting a small fraction of each population disperse between patches damps those cycles and lets both species persist. We derive a simple condition on the dispersal rate that separates persistence from collapse, and test it against simulations of up to one hundred coupled patches.
1 IntroductionEcologists have long suspected that spatial structure stabilises interacting populations [1, 2]. Classic mite experiments showed that predators and prey coexisted far longer when the habitat was divided into patches than when it was a single well-mixed arena [3]. Theory has been slower to explain why.
Most models assume that dispersal is either negligible or instantaneous. Neither describes real populations, where individuals move at finite rates and arrive at neighbouring patches already out of phase with the residents [4]. Here we treat dispersal as a continuous flow and ask how large that flow must be before the benefit appears.
2 ModelEach patch holds a prey density and a predator density. Within a patch, prey grow logistically and are consumed at a rate proportional to the product of the two densities; predators convert consumed prey into offspring and die at a constant rate [5].
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