Population models with quasi-constant-yield harvest rates

Kunquan Lan; Wei Lin

doi:10.3934/mbe.2017029

Article Contents

2017, Volume 14, Issue 2: 467-490. Doi: 10.3934/mbe.2017029

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Population models with quasi-constant-yield harvest rates

Kunquan Lan^1, , and
Wei Lin^2,

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

* Corresponding author: Kunquan Lan
* Corresponding author: Kunquan Lan

Received: August 04, 2015

Accepted: August 03, 2016

Published: September 2016

KQL was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada and by the Shanghai Key Laboratory of Contemporary Applied Mathematics, and WL was supported in part the NNSF of China under grants no. 61273014 and no. 11322111, and by the LMNS.

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Abstract

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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Mathematics Subject Classification: Primary: 34B18, 92D25; Secondary: 47H10, 47H30.

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Figure 1. (a) The areas between the $\lambda$-axis and the curve of the upper bound of $\sigma(\lambda)$ in (2.15) is a feasible region for choosing $\sigma(\lambda)$ when $h(x)\equiv\sigma(\lambda)$. (b) The area between the $\lambda$-axis and the curve of the upper bound of $\gamma$ in (2.17) is a feasible region for choosing $\gamma(\lambda)$ in Theorem 2.2

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