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. Google Scholar

[2]

V. AnuradhaD. 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. Google Scholar

[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.Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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E. ConwayR. 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. Google Scholar

[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. Google Scholar

[10]

J. Dugundji, An extension of Tietze's theorem, Pacific J. Math., 1 (1951), 353-367. doi: 10.2140/pjm.1951.1.353. Google Scholar

[11]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 353-369. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

K. Q. Lan, Multiple positive solutions of Hammerstein integral equations with singularities, Differential Equations Dynam. Systems, 8 (2000), 175-192. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

D. LudwigD. C. Aronson and H. F. Weinberger, Spatial patterning of the spruce budworm, J. Math. Biology, 8 (1979), 217-258. doi: 10.1007/BF00276310. Google Scholar

[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. Google Scholar

[22]

M. G. Neubert, Marine reserves and optimal harvesting, Ecol. Lett., 6 (2003), 843-849. Google Scholar

[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. Google Scholar

[24]

S. OrugantiJ. 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. Google Scholar

[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. Google Scholar

[26]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

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. Google Scholar

[2]

V. AnuradhaD. 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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

E. ConwayR. 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. Google Scholar

[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. Google Scholar

[10]

J. Dugundji, An extension of Tietze's theorem, Pacific J. Math., 1 (1951), 353-367. doi: 10.2140/pjm.1951.1.353. Google Scholar

[11]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 353-369. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

K. Q. Lan, Multiple positive solutions of Hammerstein integral equations with singularities, Differential Equations Dynam. Systems, 8 (2000), 175-192. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

D. LudwigD. C. Aronson and H. F. Weinberger, Spatial patterning of the spruce budworm, J. Math. Biology, 8 (1979), 217-258. doi: 10.1007/BF00276310. Google Scholar

[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. Google Scholar

[22]

M. G. Neubert, Marine reserves and optimal harvesting, Ecol. Lett., 6 (2003), 843-849. Google Scholar

[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. Google Scholar

[24]

S. OrugantiJ. 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. Google Scholar

[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. Google Scholar

[26]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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