# American Institute of Mathematical Sciences

April  2017, 14(2): 467-490. doi: 10.3934/mbe.2017029

## 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

* Corresponding author: Kunquan Lan

Received  August 05, 2015 Accepted  August 04, 2016 Published  October 2016

Fund Project: 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.

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.

Citation: Kunquan Lan, Wei Lin. Population models with quasi-constant-yield harvest rates. Mathematical Biosciences & Engineering, 2017, 14 (2) : 467-490. doi: 10.3934/mbe.2017029
##### References:
 [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM. Rev., 18 (1976), 620-709.  doi: 10.1137/1018114. [2] V. Anuradha, D. D. Hai and R. Shivaji, Existence results for superlinear semipositone BVP's, Proc. Amer. Math. Soc., 124 (1996), 757-763.  doi: 10.1090/S0002-9939-96-03256-X. [3] J. E. M. Baillie, C. Hilton-Taylor and S. N. Stuart, eds., IUCN red list of threatened species, A Global Species Assessment, IUCN, Gland, Switzerland, Cambridge, UK, 2004. [4] F. Brauer and D. A. Sánchez, Constant rate population harvesting: Equilibrium and stability, Theor. Popula. Biology, 8 (1975), 12-30.  doi: 10.1016/0040-5809(75)90036-2. [5] F. Brauer and A. C. Soudack, Coexistence properties of some predator-prey systems under constant rate harvesting and stocking, J. Math. Biol., 12 (1981), 101-114.  doi: 10.1007/BF00275206. [6] F. Brauer and A. C. Soudack, Stability regions in predator-prey systems with constant-rate prey harvesting, J. Math. Biol., 8 (1979), 55-71.  doi: 10.1007/BF00280586. [7] C. W. Clark, Mathmatics Bioeconomics, The Optimal Management of Renewable Resources Second edition, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1990. [8] E. Conway, R. Gardner and J. Smoller, Stability and bifurcation of steady-state solutions for predator-prey equations, Adv. Appl. Math., 3 (1982), 288-334.  doi: 10.1016/S0196-8858(82)80009-2. [9] E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743.  doi: 10.1090/S0002-9947-1984-0743741-4. [10] J. Dugundji, An extension of Tietze's theorem, Pacific J. Math., 1 (1951), 353-367.  doi: 10.2140/pjm.1951.1.353. [11] R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 353-369. [12] K. Q. Lan, Multiple positive solutions of semi-positone Sturm-Liouville boundary value problems, Bull. London Math. Soc., 38 (2006), 283-293.  doi: 10.1112/S0024609306018327. [13] K. Q. Lan, Positive solutions of semi-positone Hammerstein integral equations and applications, Commun. Pure Appl. Anal., 6 (2007), 441-451.  doi: 10.3934/cpaa.2007.6.441. [14] K. Q. Lan, Eigenvalues of semi-positone Hammerstein integral equations and applications to boundary value problems, Nonlinear Anal., 71 (2009), 5979-5993.  doi: 10.1016/j.na.2009.05.022. [15] K. Q. Lan, Multiple positive solutions of semilinear differential equations with singularities, J. London Math. Soc., 63 (2001), 690-704.  doi: 10.1112/S002461070100206X. [16] K. Q. Lan, Multiple positive solutions of Hammerstein integral equations with singularities, Differential Equations Dynam. Systems, 8 (2000), 175-192. [17] K. Q. Lan and C. R. Zhu, Phase portraits of predator-prey systems with harvesting rates, Discrete Contin. Dyn. Syst. Ser. A., 32 (2012), 901-933.  doi: 10.3934/dcds.2012.32.901. [18] K. Q. Lan and J. R. L. Webb, Positive solutions of semilinear differential equations with singularities, J. Differential Equations, 148 (1998), 407-421.  doi: 10.1006/jdeq.1998.3475. [19] L. G. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166.  doi: 10.1090/S0002-9947-1988-0920151-1. [20] D. Ludwig, D. C. Aronson and H. F. Weinberger, Spatial patterning of the spruce budworm, J. Math. Biology, 8 (1979), 217-258.  doi: 10.1007/BF00276310. [21] R. Ma, Positive solutions for semipositone (k, n-k) conjugate boundary value problems, J. Math. Anal. Appl., 252 (2000), 220-229.  doi: 10.1006/jmaa.2000.6987. [22] M. G. Neubert, Marine reserves and optimal harvesting, Ecol. Lett., 6 (2003), 843-849. [23] A. Okubo, Diffusion and Ecological Problems: Mathematical Models. An Extended Version of the Japanese Edition, Ecology and Diffusion Translated by G. N. Parker. Biomathematics, 10. Springer-Verlag, Berlin-New York, 1980. [24] S. Oruganti, J. Shi and R. Shivaji, Diffusive equations with constant yield harvesting, Ⅰ: Steady states, Trans. Amer. Math. Soc., 354 (2002), 3601-3619.  doi: 10.1090/S0002-9947-02-03005-2. [25] L. Roques and M. D. Chekroun, On population resilience to external perturbations, SIAM J. Appl. Math., 68 (2007), 133-153.  doi: 10.1137/060676994. [26] J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.  doi: 10.1093/biomet/38.1-2.196. [27] D. Xiao and L. S. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math., 65 (2005), 737-753.  doi: 10.1137/S0036139903428719. [28] Z. Zhao, Positive solutions of semi-positone Hammerstein integral equations and applications, Appl. Math. Comput., 219 (2012), 2789-2797.  doi: 10.1016/j.amc.2012.09.009. [29] C. R. Zhu and K. Q. Lan, Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 289-306.  doi: 10.3934/dcdsb.2010.14.289.

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##### References:
 [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM. Rev., 18 (1976), 620-709.  doi: 10.1137/1018114. [2] V. Anuradha, D. D. Hai and R. Shivaji, Existence results for superlinear semipositone BVP's, Proc. Amer. Math. Soc., 124 (1996), 757-763.  doi: 10.1090/S0002-9939-96-03256-X. [3] J. E. M. Baillie, C. Hilton-Taylor and S. N. Stuart, eds., IUCN red list of threatened species, A Global Species Assessment, IUCN, Gland, Switzerland, Cambridge, UK, 2004. [4] F. Brauer and D. A. Sánchez, Constant rate population harvesting: Equilibrium and stability, Theor. Popula. Biology, 8 (1975), 12-30.  doi: 10.1016/0040-5809(75)90036-2. [5] F. Brauer and A. C. Soudack, Coexistence properties of some predator-prey systems under constant rate harvesting and stocking, J. Math. Biol., 12 (1981), 101-114.  doi: 10.1007/BF00275206. [6] F. Brauer and A. C. Soudack, Stability regions in predator-prey systems with constant-rate prey harvesting, J. Math. Biol., 8 (1979), 55-71.  doi: 10.1007/BF00280586. [7] C. W. Clark, Mathmatics Bioeconomics, The Optimal Management of Renewable Resources Second edition, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1990. [8] E. Conway, R. Gardner and J. Smoller, Stability and bifurcation of steady-state solutions for predator-prey equations, Adv. Appl. Math., 3 (1982), 288-334.  doi: 10.1016/S0196-8858(82)80009-2. [9] E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743.  doi: 10.1090/S0002-9947-1984-0743741-4. [10] J. Dugundji, An extension of Tietze's theorem, Pacific J. Math., 1 (1951), 353-367.  doi: 10.2140/pjm.1951.1.353. [11] R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 353-369. [12] K. Q. Lan, Multiple positive solutions of semi-positone Sturm-Liouville boundary value problems, Bull. London Math. Soc., 38 (2006), 283-293.  doi: 10.1112/S0024609306018327. [13] K. Q. Lan, Positive solutions of semi-positone Hammerstein integral equations and applications, Commun. Pure Appl. Anal., 6 (2007), 441-451.  doi: 10.3934/cpaa.2007.6.441. [14] K. Q. Lan, Eigenvalues of semi-positone Hammerstein integral equations and applications to boundary value problems, Nonlinear Anal., 71 (2009), 5979-5993.  doi: 10.1016/j.na.2009.05.022. [15] K. Q. Lan, Multiple positive solutions of semilinear differential equations with singularities, J. London Math. Soc., 63 (2001), 690-704.  doi: 10.1112/S002461070100206X. [16] K. Q. Lan, Multiple positive solutions of Hammerstein integral equations with singularities, Differential Equations Dynam. Systems, 8 (2000), 175-192. [17] K. Q. Lan and C. R. Zhu, Phase portraits of predator-prey systems with harvesting rates, Discrete Contin. Dyn. Syst. Ser. A., 32 (2012), 901-933.  doi: 10.3934/dcds.2012.32.901. [18] K. Q. Lan and J. R. L. Webb, Positive solutions of semilinear differential equations with singularities, J. Differential Equations, 148 (1998), 407-421.  doi: 10.1006/jdeq.1998.3475. [19] L. G. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166.  doi: 10.1090/S0002-9947-1988-0920151-1. [20] D. Ludwig, D. C. Aronson and H. F. Weinberger, Spatial patterning of the spruce budworm, J. Math. Biology, 8 (1979), 217-258.  doi: 10.1007/BF00276310. [21] R. Ma, Positive solutions for semipositone (k, n-k) conjugate boundary value problems, J. Math. Anal. Appl., 252 (2000), 220-229.  doi: 10.1006/jmaa.2000.6987. [22] M. G. Neubert, Marine reserves and optimal harvesting, Ecol. Lett., 6 (2003), 843-849. [23] A. Okubo, Diffusion and Ecological Problems: Mathematical Models. An Extended Version of the Japanese Edition, Ecology and Diffusion Translated by G. N. Parker. Biomathematics, 10. Springer-Verlag, Berlin-New York, 1980. [24] S. Oruganti, J. Shi and R. Shivaji, Diffusive equations with constant yield harvesting, Ⅰ: Steady states, Trans. Amer. Math. Soc., 354 (2002), 3601-3619.  doi: 10.1090/S0002-9947-02-03005-2. [25] L. Roques and M. D. Chekroun, On population resilience to external perturbations, SIAM J. Appl. Math., 68 (2007), 133-153.  doi: 10.1137/060676994. [26] J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.  doi: 10.1093/biomet/38.1-2.196. [27] D. Xiao and L. S. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math., 65 (2005), 737-753.  doi: 10.1137/S0036139903428719. [28] Z. Zhao, Positive solutions of semi-positone Hammerstein integral equations and applications, Appl. Math. Comput., 219 (2012), 2789-2797.  doi: 10.1016/j.amc.2012.09.009. [29] C. R. Zhu and K. Q. Lan, Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 289-306.  doi: 10.3934/dcdsb.2010.14.289.
(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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