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Allee effect and bistability in a spatially heterogeneous predator-prey model

Authors: Yihong Du and Junping Shi
Journal: Trans. Amer. Math. Soc. 359 (2007), 4557-4593
MSC (2000): Primary 35J55, 92D40; Secondary 35B30, 35B32, 35J65, 92D25
Published electronically: April 17, 2007
MathSciNet review: 2309198
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Abstract: A spatially heterogeneous reaction-diffusion system modelling predator-prey interaction is studied, where the interaction is governed by a Holling type II functional response. Existence of multiple positive steady states and global bifurcation branches are examined as well as related dynamical behavior. It is found that while the predator population is not far from a constant level, the prey population could be extinguished, persist or blow up depending on the initial population distributions, the various parameters in the system, and the heterogeneous environment. In particular, our results show that when the prey growth is strong, the spatial heterogeneity of the environment can play a dominant role for the presence of the Allee effect. Our mathematical analysis relies on bifurcation theory, topological methods, various comparison principles and elliptic estimates. We combine these methods with monotonicity arguments to the system through the use of some new auxiliary scalar equations, though the system itself does not keep an order structure as the competition system does. Among other things, this allows us to obtain partial descriptions of the dynamical behavior of the system.

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  • [BL] Berestycki, H.; Lions, P.-L., Some applications of the method of super and subsolutions. Lecture Notes in Math., 782, Springer, Berlin, 1980, pp. 16-41. MR 572249 (82c:35021)
  • [BNV] Berestycki, H.; Nirenberg, L.; Varadhan, S.R.S., The principle eigenvalue and maximum principle for second-order elliptic operators in general domains. Comm. Pure Appl. Math. 47 (1994), no. 1, 47-92. MR 1258192 (95h:35053)
  • [BB] Blat, J.; Brown, K.J., Global bifurcation of positive solutions in some systems of elliptic equations. SIAM J. Math. Anal., 17 (1986), 1339-1353. MR 860917 (87k:35077)
  • [BD] Brown, K. J.; Du, Yihong, Bifurcation and monotonicity in competition reaction-diffusion systems. Nonlinear Anal., 23 (1994), 1-13. MR 1288495 (95h:35112)
  • [CC1] Cantrell, Robert Stephen; Cosner, Chris, Diffusive logistic equations with indefinite weights: Population models in disrupted environments. Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), no. 3-4, 293-318. MR 1014659 (91b:92015)
  • [CC2] Cantrell, Robert Stephen; Cosner, Chris, Spatial ecology via reaction-diffusion equations. Wiley Series in Mathematical and Computational Biology, John Wiley & Sons Ltd., 2003. MR 2191264
  • [CEL] Casal, A.; Eilbeck, J. C.; López-Gómez, J. Existence and uniqueness of coexistence states for a predator-prey model with diffusion. Differential Integral Equations 7 (1994), no. 2, 411-439. MR 1255897 (95f:92009)
  • [CP] Clément, Ph.; Peletier, L. A., An anti-maximum principle for second-order elliptic operators. J. Differential Equations, 34, (1979), no. 2, 218-229. MR 550042 (83c:35034)
  • [Co] Conway, E. D., Diffusion and predator-prey interaction: pattern in closed systems. Partial differential equations and dynamical systems, 85-133, Res. Notes in Math., 101, Pitman, Boston, Mass.-London, (1984). MR 759745 (86e:92035)
  • [CR1] Crandall, Michael G.; Rabinowitz, Paul H., Bifurcation from simple eigenvalues. J. Functional Analysis, 8 (1971), 321-340. MR 0288640 (44:5836)
  • [CR2] Crandall, Michael G.; Rabinowitz, Paul H., Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Rational Mech. Anal. 52 (1973), 161-180. MR 0341212 (49:5962)
  • [Da1] Dancer, E. N., On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151. MR 688538 (84d:58020)
  • [Da2] Dancer, E. N., On positive solutions of some pairs of differential equations. Trans. Amer. Math. Soc. 284 (1984), no. 2, 729-743. MR 743741 (85i:35056)
  • [DD] Dancer, E. N.; Du, Yihong, Effects of certain degeneracies in the predator-prey model. SIAM J. Math. Anal. 34 (2002), no. 2, 292-314. MR 1951776 (2004a:35064)
  • [dMR] de Mottoni, P.; Rothe, F., Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems with diffusion. SIAM J. Appl. Math. 37(1979), 648-663. MR 549146 (81b:92008)
  • [D1] Du, Yihong, Effects of a degeneracy in the competition model. I. Classical and generalized steady-state solutions. J. Differential Equations 181 (2002), no. 1, 92-132. MR 1900462 (2003d:35125)
  • [D2] Du, Yihong, Effects of a degeneracy in the competition model. II. Perturbation and dynamical behaviour. J. Differential Equations 181 (2002), no. 1, 133-164. MR 1900463 (2003d:35124)
  • [D3] Du, Yihong, Realization of prescribed patterns in the competition model. J. Differential Equations 193 (2003), no. 1, 147-179. MR 1994062 (2004j:35095)
  • [D4] Du, Yihong, Spatial patterns for population models in a heterogeneous environment. Taiwanese J. Math. 8 (2004), no. 2, 155-182. MR 2061685 (2005e:92035)
  • [D5] Du, Yihong, Bifurcation and related topics in elliptic problems. Stationary partial differential equations. Vol. II, 127-209, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005. MR 2181483 (2006k:35113)
  • [D6] Du, Yihong, Asymptotic behavior and uniqueness results for boundary blow-up solutions. Diff. Integral Eqns., 17(2004), 819-834. MR 2074688 (2005f:35096)
  • [DHs] Du, Yihong; Hsu, Sze-Bi, A diffusive predator-prey model in heterogeneous environment. J. Differential Equations 203 (2004), no. 2, 331-364. MR 2073690 (2005e:35051)
  • [DHu] Du, Yihong; Huang, Qingguang, Blow-up solutions for a class of semilinear elliptic and parabolic equations. SIAM J. Math. Anal. 31 (1999), no. 1, 1-18. MR 1720128 (2000g:35059)
  • [DLi] Du, Yihong; Li, Shujie, Positive solutions with prescribed patterns in some simple semilinear equations. Differential Integral Equations 15 (2002), no. 7, 805-822. MR 1895567 (2003f:35105)
  • [DL1] Du, Yihong; Lou, Yuan, Some uniqueness and exact multiplicity results for a predator-prey model. Trans. Amer. Math. Soc. 349 (1997), no. 6, 2443-2475. MR 1401768 (98e:35164)
  • [DL2] Du, Yihong; Lou, Yuan, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model. J. Differential Equations 144 (1998), no. 2, 390-440. MR 1616901 (99e:35046)
  • [DL3] Du, Yihong; Lou, Yuan, Qualitative behaviour of positive solutions of a predator-prey model: Effects of saturation. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 2, 321-349. MR 1830414 (2002e:35125)
  • [DM] Du, Yihong; Ma, Li, Logistic type equations on $ \mathbf{R}\sp N$ by a squeezing method involving boundary blow-up solutions. J. London Math. Soc. (2) 64 (2001), no. 1, 107-124. MR 1840774 (2002d:35089)
  • [DS1] Du, Yihong; Shi, Junping, Spatially Heterogeneous Predator-Prey Model with Protect Zone for Prey. Jour. Diff. Equa., 229 (2006), no. 1, 63-91. MR 2265618
  • [DS2] Du, Yihong; Shi, Junping, Some Recent Results on Diffusive Predator-prey Models in Spatially Heterogeneous Environment. Nonlinear dynamics and evolution equations, Fields Institute Communications, 48, American Mathematical Society (2006), pp. 95-135. MR 2223350
  • [H] Holling, C. S., Some characteristics of simple types of predation and parasitism. Canadian Entomologist 91 (1959), 385-398.
  • [HLR] Huang, Jianhua; Lu, Gang; Ruan, Shigui, Existence of traveling wave solutions in a diffusive predator-prey model. J. Math. Biol. 46 (2003), no. 2, 132-152. MR 1963069 (2004b:35183)
  • [KS] Korman, Philip; Shi, Junping, New exact multiplicity results with an application to a population model. Proc. Royal Soc. Edin. A, 131 (2001), No. 5, 1167-1182. MR 1862448 (2002h:35092)
  • [KuS] Kurata, Kazuhiro; Shi, Junping, Optimal spatial harvesting strategy and symmetry-breaking. Submitted, (2005).
  • [MP] Medvinsky, Alexander B.; Petrovskii, Sergei V.; Tikhonova, Irene A.; Malchow, Horst; Li, Bai-Lian, Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev. 44 (2002), no. 3, 311-370. MR 1951363 (2003m:92083)
  • [OSS] Oruganti, Shobha, Shi, Junping; Shivaji, Ratnasingham, Diffusive logistic equation with constant yield harvesting, I: Steady States. Trans. Amer. Math. Soc., 354, (2002), no. 9, 3601-3619,. MR 1911513 (2003g:35080)
  • [O] Ouyang, Tiancheng, On the positive solutions of semilinear equations $ \Delta u+\lambda u-hu\sp p=0$ on the compact manifolds. Trans. Amer. Math. Soc. 331 (1992), no. 2, 503-527. MR 1055810 (92h:35012)
  • [OS1] Ouyang, Tiancheng; Shi, Junping, Exact multiplicity of positive solutions for a class of semilinear problems. Jour. Diff. Equa. 146 (1998), no. 1, 121-156. MR 1625731 (99f:35061)
  • [OS2] Ouyang, Tiancheng; Shi, Junping, Exact multiplicity of positive solutions for a class of semilinear problem: II. Jour. Diff. Equa. 158, (1999), no. 1, 94-151. MR 1721723 (2001b:35117)
  • [OL] Owen, M.R.; Lewis, M.A., How can predation slow, stall or reverse a prey invasion? Bull. Math. Biol. 63, (2001), 655-684.
  • [SJ] Segel, L.A.; Jackson, J.L., Dissipative structure: An explaination and an ecological example. J. Theor. Biol., 37, (1975), 545-559.
  • [SL] Segel, Lee A.; Levin, Simon A. Application of nonlinear stability theory to the study of the effects of diffusion on predator-prey interactions. AIP Conf. Proc., 27, Amer. Inst. Phys., New York, (1976), 123-156.. MR 0496857 (58:15316)
  • [S1] Shi, Junping, Persistence and bifurcation of degenerate solutions. Jour. Funct. Anal. 169, (1999), No. 2, 494-531. MR 1730558 (2001h:47115)
  • [S2] Shi, Junping, A radially symmetric anti-maximum principle and applications to fishery management models. Electronic Journal of Differential Equations. 2004, No. 27, 1-13, (2004). MR 2036211 (2004k:34031)
  • [SS1] Shi, Junping; Shivaji, Ratnasingham, Global bifurcations of concave semipositone problems. Advances in Evolution Equations: Proceedings in honor of J. A. Goldstein's 60th birthday, Edited by G.R. Goldstein, R. Nagel, and S. Romanelli, Marcel Dekker, Inc., New York, Basel, 385-398, (2003). MR 2073759 (2005g:35113)
  • [SS2] Shi, Junping; Shivaji, Ratnasingham, Persistence in reaction diffusion models with weak Allee effect. Jour. Math. Biol. 52 (2006), 807-829. MR 2235529
  • [SY] Shi, Junping; Yao, Miaoxin, On a singular nonlinear semilinear elliptic problem. Proc. Royal Soc. Edin. A, 128, (1998), no. 6, 1389-1401. MR 1663988 (99j:35070)
  • [Si] Simon, Leon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems. Ann. of Math. 118 (1983), no. 3, 525-571. MR 727703 (85b:58121)

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Additional Information

Yihong Du
Affiliation: School of Mathematics, Statistics and Computer Sciences, University of New England, Armidale, NSW2351, Australia – and – Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, People’s Republic of China

Junping Shi
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187 – and – School of Mathematics, Harbin Normal University, Harbin, Heilongjiang 150025, People’s Republic of China

Keywords: Reaction-diffusion system, predator-prey model, spatial heterogeneity.
Received by editor(s): April 6, 2005
Received by editor(s) in revised form: February 10, 2006
Published electronically: April 17, 2007
Additional Notes: The first author was partially supported by the Australia Research Council
The second author was partially supported by United States NSF grants DMS-0314736 and EF-0436318, College of William and Mary junior research leave, and a grant from Science Council of Heilongjiang Province, China.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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