Non-local Dispersal Models for a Population under Climate Change (Joy) Ying Zhou, Mark Kot Department of Applied Mathematics University of Washington 1

Cartoon of a Range Shift 2

3 Global mean: 0.42km/yr

Cartoon of a Range Shift 4 Population Dynamics Matter

Talk Outline 5 Population Models on Range Shifts under: Constant-speed climate change Accelerated climate change

Organisms of Interest Well-defined life stages (growth, dispersal) Growth and dispersal occur in separate time periods Non-overlapping generations Larvae Adult Egg mass Flower Seed Seedling Cocoon

Integrodifference equation 7 Integrodifference eqn (IDE) kernel Assuming no Allee effects

How To Mathematize Climate Warming? 8

Climatically Suitable Habitat Habitat shifts 9 Combination of two classical problems Zhou and Kot 2011 Theoretical Ecology

Two Classic IDE Models 10

Two Classic IDE Models 11

What Population Dynamics Will We Observe? A Steady Range Shift For Small c 12 Zhou and Kot 2011 Theoretical Ecology

Extinction When c Large 13 Zhou and Kot 2011 Theoretical Ecology

Critical Speed “c*” Viability of a population Ability to establish itself at a low density Instability of the trivial equilibrium Dominant eigenvalue of an integral operator exceeding 1 14

Eigenvalue Problem Net reproductive rate Analytic method for “separable” kernels Numerical method “Nystrom’s method” Delves and Wash 1974

Larger Net Reproductive Rate Helps 16 Zhou and Kot 2011 Theoretical Ecology

More Dispersal, But Not Over-dispersal 17 Dispersal radius radius Zhou and Kot 2011 Theoretical Ecology

18 Lockwood et al. 2002

Clark 1998 Mean deviation 19 Schultz 1998

Result for a typical leptokurtic kernel The “Tail” of The Dispersal Kernel Result for a typical leptokurtic kernel Result for a typical platykurtic kernel 20 Zhou and Kot 2011 Theoretical Ecology

Population projection matrix Matrix of dispersal kernels Vector of population density in each stage

Climatically Suitable Habitat Habitat shifts Heterogeneous Habitat Suitability 22 Habitat quality function Latore et al. 1999

Consider linearized equation For normally distributed habitat quality a Gaussian dispersal kernel

and a special initial condition (Gaussian initial profile), then we have an ansatz : peak of the pulse : amplitude of the pulse Latore et al. 1999

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26 “climate deficit”

27 Declining population if

Accelerated Climate Change Same ansatz

The mean of the Gaussian ansatz

30 The “climate deficit”

Time Speed T

32 vs. For large t Comparison of climate deficit

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Summary An integrodifference equation model with shifting boundaries Critical speed Acceleration may hurt a lot (more than average) 35

Thank you! Questions? 36