Bifurcation curves of a logistic equation when the linear growth rate crosses a second eigenvalue

Pedro M. Girão

doi:10.1016/j.na.2010.08.020

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arXiv · 1406.7415

Bifurcation curves of a logistic equation when the linear growth rate crosses a second eigenvalue

Pedro M. Girão

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Abstract

We construct the global bifurcation curves, solutions versus level of harvesting, for the steady states of a diffusive logistic equation on a bounded domain, under Dirichlet boundary conditions and other appropriate hypotheses, when $a$, the linear growth rate of the population, is below $\lambda_2+\delta$. Here $\lambda_2$ is the second eigenvalue of the Dirichlet Laplacian on the domain and $\delta>0$. Such curves have been obtained before, but only for $a$ in a right neighborhood of the first eigenvalue. Our analysis provides the exact number of solutions of the equation for $a\leq\lambda_2$ and new information on the number of solutions for $a>\lambda_2$.

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Pedro M. Girão. 2014-06-28. Bifurcation curves of a logistic equation when the linear growth rate crosses a second eigenvalue. https://doi.org/10.1016/j.na.2010.08.020

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