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Population models with quasi-constant-yield harvest rates

1. Department of Mathematics, Ryerson University, Toronto, Ontario, M5B 2K3, Canada
2. School of Mathematical Sciences and Centre for Computational Systems Biology, Fudan University, Shanghai 200433, China

One-dimensional logistic population models with quasi-constant-yield harvest rates are studied under the assumptions that a population inhabits a patch of dimensionless width and no members of the population can survive outside of the patch. The essential problem is to determine the size of the patch and the ranges of the harvesting rate functions under which the population survives or becomes extinct. This is the first paper which discusses such models with the Dirichlet boundary conditions and can tell the exact quantity of harvest rates of the species without having the population die out. The methodology is to establish new results on the existence of positive solutions of semi-positone Hammerstein integral equations using the fixed point index theory for compact maps defined on cones, and apply the new results to tackle the essential problem. It is expected that the established analytical results have broad applications in management of sustainable ecological systems.

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Keywords Logistic population model; quasi-constant-yield harvest rate; Dirichlet boundary condition; semi-positone integral equation; fixed point index

Citation: Kunquan Lan, Wei Lin. Population models with quasi-constant-yield harvest rates. Mathematical Biosciences and Engineering, 2017, 14(2): 467-490. doi: 10.3934/mbe.2017029


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