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November 2018, 23(9): 3755-3786. doi: 10.3934/dcdsb.2018076

Extinction and coexistence of species for a diffusive intraguild predation model with B-D functional response

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

* Corresponding author

Received  January 2016 Revised  September 2017 Published  March 2018

Fund Project: This work is supported by National Science Foundation of China(11701472), Fundamental Research Funds for the Central Universities(XDJK2016C121, XDJK2017C055), the Ph.D. Foundation of Southwest University (SWU112099, SWU116069)

Extinction and coexistence of species are two fundamental issues in systems with IGP. In this paper, we constructed a mathematical model with IGP by introducing heterogeneous environment and B-D functional response between the predator and prey. First, some sufficient conditions for the extinction and permanence of the time-dependent system were obtained by using comparison principle and upper and lower solution method. Second, we got some necessary and sufficient conditions for the existence of coexistence states by means of the fixed point index theory. In addition, we discussed the uniqueness and stability of coexistence state under some conditions. Finally, we studied the effects of the parameters in system on the spatial distribution of species and obtained some interesting results about the extinction and coexistence of species by using numerical simulations.

Citation: Guohong Zhang, Xiaoli Wang. Extinction and coexistence of species for a diffusive intraguild predation model with B-D functional response. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3755-3786. doi: 10.3934/dcdsb.2018076
References:
[1]

P. A. Abrams and S. R. Fung, Prey persistence and abundance in systems with intraguild predation and type-2 functional responses, J. Theor. Biol., 264 (2010), 1033-1042. doi: 10.1016/j.jtbi.2010.02.045.

[2]

P. Amarasekare, Spatial dynamics of communities with intraguild predation: The role of dispersal strategies, The American Naturalist, 170 (2007), 819-831. doi: 10.1086/522837.

[3]

P. Amarasekare, Coexistence of intraguild predators and prey in resource-rich environments, Ecology, 89 (2008), 2786-2797. doi: 10.1890/07-1508.1.

[4]

M. Arim and P. A. Marquet, Intraguild predation: A wide spread interaction related to species biology, Ecology Letters, 7 (2004), 557-564. doi: 10.1111/j.1461-0248.2004.00613.x.

[5]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, Journal of Animal Ecology, 44 (1975), 331-340. doi: 10.2307/3866.

[6]

J. Blat and K. J. Brown, Global bifurcation of positive solutions in some elliptic systems of elliptic equations, SIAM J. Math. Anal., 17 (1986), 1339-1353. doi: 10.1137/0517094.

[7]

S. Cano-Casanova, Existence and structure of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems, Nonlinear Anal., 49 (2002), 361-430. doi: 10.1016/S0362-546X(01)00116-X.

[8]

S. Cano-Casanova and J. López-Gómez, Properties of the principle eigenvalues of a general class of non-classical mixed boundary value problems, J. Differential Equations, 178 (2002), 123-211. doi: 10.1006/jdeq.2000.4003.

[9]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations Wiley, Chichester, UK, 2003.

[10]

J. A. Collera, Bifurcations in delayed Lotka-Volterra intraguild predation model, Journal of the Mathematical Society of the Philippines, 37 (2014), 11-22.

[11]

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.

[12]

E. N. Dancer, On positive solutions of some pairs of differential equations-Ⅱ, J. Differential Equations, 60 (1985), 236-258. doi: 10.1016/0022-0396(85)90115-9.

[13]

E. N. Dancer and Y. H. Du, Positive solutions for a three-species competition system with diffusion-Ⅰ. General existence results, Nonlinear Anal., 24 (1995), 337-357. doi: 10.1016/0362-546X(94)E0063-M.

[14]

D. L. DeAngelisR. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892.

[15]

M. FreezeY. Chang and M. Feng, Analysis of dynamics in a complex food chain with ratio-dependent functional response, J. Appl. Anal. Comput., 4 (2014), 69-87.

[16]

G. Guo and J. Wu, Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response, Nonlinear Anal., 72 (2010), 1632-1646. doi: 10.1016/j.na.2009.09.003.

[17]

R. D. Holt and G. Huxel, Alternative prey and the dynamics of intraguild predation:theoretical perspectives, Ecology, 88 (2007), 2706-2712.

[18]

R. D. Holt and G. A. Polis, A theoretical framework for intraguild predation, The American Naturalist, 149 (1997), 745-764. doi: 10.1086/286018.

[19]

S. B. HsuS. Ruan and T. H. Yang, Analysis of three species Lotka-Volterra food web models with omnivory, J. Math. Anal. Appl., 426 (2015), 659-687. doi: 10.1016/j.jmaa.2015.01.035.

[20]

Y. IkegawaH. Ezoe and T. Namba, Adaptive defense of pests and switching predation can improve biological control by multiple natural enemies, Population Ecology, 57 (2015), 381-395. doi: 10.1007/s10144-014-0468-8.

[21]

C. JamesV. LecomteaY. Dumontb and M. L. Correa, Intraguild predation and mesopredator release effect on long-lived prey, Ecological Modelling, 220 (2009), 1098-1104. doi: 10.1016/j.ecolmodel.2009.01.017.

[22]

Y. Kang and L. Wedekin, Dynamics of a intraguild predation model with generalist or specialist predator, J. Math. Biol., 67 (2013), 1227-1259. doi: 10.1007/s00285-012-0584-z.

[23]

T. KimbrellR. D. Holt and P. Lundberg, The infuence of vigilance on intraguild predation, J. Theor. Biol., 249 (2007), 218-234. doi: 10.1016/j.jtbi.2007.07.031.

[24]

W. Ko and K. Ryu, Analysis of diffusive two-competing-prey and one-predator systems with Beddington-Deangelis functional response, Nonlinear Anal., 71 (2009), 4185-4202. doi: 10.1016/j.na.2009.02.119.

[25]

M. A. Krasnoselskii, Positive Solutions of Operator Equations Noordhoff, Groningen, 1964.

[26]

V. Křivan and S. Diehl, Adaptive omnivory and species coexistence in tri-trophicfood webs, Theor. Popul. Biol., 67 (2005), 85-99. doi: 10.1016/j.tpb.2004.09.003.

[27]

L. D. J. KuijperB. W. KooiC. Zonneveld and S. A. L. M. Kooijman, Omnivory and food web dynamics, Ecological Modelling, 163 (2003), 19-32. doi: 10.1016/S0304-3800(02)00351-4.

[28]

L. 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.

[29]

Z. Liu and F. Zhang, Species coexistence of communities with intraguild predation: Therole of refuges used by the resource and the intraguild prey, BioSystems, 114 (2013), 25-30. doi: 10.1016/j.biosystems.2013.07.010.

[30]

J. López-Gómez and R. Pardo, Coexistence regions in Lotka-Volterra models with diffusion, Nonlinear Anal., 19 (1992), 11-28. doi: 10.1016/0362-546X(92)90027-C.

[31]

T. NambaK. Tanabe and N. Maeda, Omnivory and stability of food webs, Ecological Complexity, 5 (2008), 73-85. doi: 10.1016/j.ecocom.2008.02.001.

[32]

T. Okuyama and R. L. Ruyle, Analysis of adaptive foraging in an intraguild predation system, Web Ecology, 4 (2003), 1-6. doi: 10.5194/we-4-1-2003.

[33]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations Plenum Press, New York, 1992.

[34]

C. V. Pao, Quasisolutions and global attractor of reaction-diffusion systems, Nonlinear Anal., 26 (1996), 1889-1903. doi: 10.1016/0362-546X(95)00058-4.

[35]

G. A. PolisC. A. Myers and R. D. Holt, The ecology and evolution of intraguild predation: Potential competitors that eat each other, Annual Review of Ecology and Systematics, 20 (1989), 297-330. doi: 10.1146/annurev.es.20.110189.001501.

[36]

G. A. Polis and R. D. Holt, Intraguild predation: the dynamics of complex trophic interactions, Trends in Ecology and Evolution, 7 (1992), 151-154. doi: 10.1016/0169-5347(92)90208-S.

[37]

D. Ryan and R. S. Cantrell, Aoidance behavior in intraguild predation communities: A cross-diffusion model, Discrete Contin. Dyn. Syst. A, 35 (2015), 1641-1663.

[38]

T. Schellekens and T. V. Kooten, Coexistence of two stage-structured intraguild predators, J. Theor. Biol., 308 (2012), 36-44. doi: 10.1016/j.jtbi.2012.05.017.

[39]

H. ShuX. HuL. Wang and J. Watmough, Delay induced stability switch, multitype bistability and chaos in an intraguild predation model, J. Math. Biol., 71 (2015), 1269-1298. doi: 10.1007/s00285-015-0857-4.

[40]

J. Smoller, Shock Waves and Reaction-Diffusion Equations Second edition, in Grundkehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), 258, Spring-Verlag New York, 1994.

[41]

K. Tanabe and T. Namba, Omnivory creates chaos in simple food webmodels, Ecology, 86 (2005), 3411-3414.

[42]

P. Urbania and R. R. Jiliberto, Adaptive prey behavior and the dynamics of intraguild predation systems, Ecological Modelling, 221 (2010), 2628-2633. doi: 10.1016/j.ecolmodel.2010.08.009.

[43]

A. Verdy and P. Amarasekare, Alternative stable states in communities with intraguild predation, J. Theor. Biol., 262 (2010), 116-128. doi: 10.1016/j.jtbi.2009.09.011.

[44]

A. W. VisserP. Mariani and S. Pigolotti, Adaptive behaviour, tri-trophic foodweb stability and damping of chaos, Journal of the Royal Society Interface, 9 (2012), 1373-1380. doi: 10.1098/rsif.2011.0686.

[45]

M. YamaguchiY. Takeuchi and W. Ma, Dynamical properties of a stage structured three-species model with intra-guild predation, Journal of Computational and Applied Mathematics, 201 (2007), 327-338. doi: 10.1016/j.cam.2005.12.033.

[46]

G. ZhangW. Wang and X. Wang, Coexistence states for a diffusive one-prey and two-predators model with B-D functional response, J. Math. Anal. Appl., 387 (2012), 931-948. doi: 10.1016/j.jmaa.2011.09.049.

[47]

J. Zhou and C. Mu, Positive solutions for a three-trophic food chain model with diffusion and Beddington-Deangelis functional response, Nonlinear Anal. Real World Appl, 12 (2011), 902-917. doi: 10.1016/j.nonrwa.2010.08.015.

show all references

References:
[1]

P. A. Abrams and S. R. Fung, Prey persistence and abundance in systems with intraguild predation and type-2 functional responses, J. Theor. Biol., 264 (2010), 1033-1042. doi: 10.1016/j.jtbi.2010.02.045.

[2]

P. Amarasekare, Spatial dynamics of communities with intraguild predation: The role of dispersal strategies, The American Naturalist, 170 (2007), 819-831. doi: 10.1086/522837.

[3]

P. Amarasekare, Coexistence of intraguild predators and prey in resource-rich environments, Ecology, 89 (2008), 2786-2797. doi: 10.1890/07-1508.1.

[4]

M. Arim and P. A. Marquet, Intraguild predation: A wide spread interaction related to species biology, Ecology Letters, 7 (2004), 557-564. doi: 10.1111/j.1461-0248.2004.00613.x.

[5]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, Journal of Animal Ecology, 44 (1975), 331-340. doi: 10.2307/3866.

[6]

J. Blat and K. J. Brown, Global bifurcation of positive solutions in some elliptic systems of elliptic equations, SIAM J. Math. Anal., 17 (1986), 1339-1353. doi: 10.1137/0517094.

[7]

S. Cano-Casanova, Existence and structure of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems, Nonlinear Anal., 49 (2002), 361-430. doi: 10.1016/S0362-546X(01)00116-X.

[8]

S. Cano-Casanova and J. López-Gómez, Properties of the principle eigenvalues of a general class of non-classical mixed boundary value problems, J. Differential Equations, 178 (2002), 123-211. doi: 10.1006/jdeq.2000.4003.

[9]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations Wiley, Chichester, UK, 2003.

[10]

J. A. Collera, Bifurcations in delayed Lotka-Volterra intraguild predation model, Journal of the Mathematical Society of the Philippines, 37 (2014), 11-22.

[11]

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.

[12]

E. N. Dancer, On positive solutions of some pairs of differential equations-Ⅱ, J. Differential Equations, 60 (1985), 236-258. doi: 10.1016/0022-0396(85)90115-9.

[13]

E. N. Dancer and Y. H. Du, Positive solutions for a three-species competition system with diffusion-Ⅰ. General existence results, Nonlinear Anal., 24 (1995), 337-357. doi: 10.1016/0362-546X(94)E0063-M.

[14]

D. L. DeAngelisR. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892.

[15]

M. FreezeY. Chang and M. Feng, Analysis of dynamics in a complex food chain with ratio-dependent functional response, J. Appl. Anal. Comput., 4 (2014), 69-87.

[16]

G. Guo and J. Wu, Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response, Nonlinear Anal., 72 (2010), 1632-1646. doi: 10.1016/j.na.2009.09.003.

[17]

R. D. Holt and G. Huxel, Alternative prey and the dynamics of intraguild predation:theoretical perspectives, Ecology, 88 (2007), 2706-2712.

[18]

R. D. Holt and G. A. Polis, A theoretical framework for intraguild predation, The American Naturalist, 149 (1997), 745-764. doi: 10.1086/286018.

[19]

S. B. HsuS. Ruan and T. H. Yang, Analysis of three species Lotka-Volterra food web models with omnivory, J. Math. Anal. Appl., 426 (2015), 659-687. doi: 10.1016/j.jmaa.2015.01.035.

[20]

Y. IkegawaH. Ezoe and T. Namba, Adaptive defense of pests and switching predation can improve biological control by multiple natural enemies, Population Ecology, 57 (2015), 381-395. doi: 10.1007/s10144-014-0468-8.

[21]

C. JamesV. LecomteaY. Dumontb and M. L. Correa, Intraguild predation and mesopredator release effect on long-lived prey, Ecological Modelling, 220 (2009), 1098-1104. doi: 10.1016/j.ecolmodel.2009.01.017.

[22]

Y. Kang and L. Wedekin, Dynamics of a intraguild predation model with generalist or specialist predator, J. Math. Biol., 67 (2013), 1227-1259. doi: 10.1007/s00285-012-0584-z.

[23]

T. KimbrellR. D. Holt and P. Lundberg, The infuence of vigilance on intraguild predation, J. Theor. Biol., 249 (2007), 218-234. doi: 10.1016/j.jtbi.2007.07.031.

[24]

W. Ko and K. Ryu, Analysis of diffusive two-competing-prey and one-predator systems with Beddington-Deangelis functional response, Nonlinear Anal., 71 (2009), 4185-4202. doi: 10.1016/j.na.2009.02.119.

[25]

M. A. Krasnoselskii, Positive Solutions of Operator Equations Noordhoff, Groningen, 1964.

[26]

V. Křivan and S. Diehl, Adaptive omnivory and species coexistence in tri-trophicfood webs, Theor. Popul. Biol., 67 (2005), 85-99. doi: 10.1016/j.tpb.2004.09.003.

[27]

L. D. J. KuijperB. W. KooiC. Zonneveld and S. A. L. M. Kooijman, Omnivory and food web dynamics, Ecological Modelling, 163 (2003), 19-32. doi: 10.1016/S0304-3800(02)00351-4.

[28]

L. 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.

[29]

Z. Liu and F. Zhang, Species coexistence of communities with intraguild predation: Therole of refuges used by the resource and the intraguild prey, BioSystems, 114 (2013), 25-30. doi: 10.1016/j.biosystems.2013.07.010.

[30]

J. López-Gómez and R. Pardo, Coexistence regions in Lotka-Volterra models with diffusion, Nonlinear Anal., 19 (1992), 11-28. doi: 10.1016/0362-546X(92)90027-C.

[31]

T. NambaK. Tanabe and N. Maeda, Omnivory and stability of food webs, Ecological Complexity, 5 (2008), 73-85. doi: 10.1016/j.ecocom.2008.02.001.

[32]

T. Okuyama and R. L. Ruyle, Analysis of adaptive foraging in an intraguild predation system, Web Ecology, 4 (2003), 1-6. doi: 10.5194/we-4-1-2003.

[33]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations Plenum Press, New York, 1992.

[34]

C. V. Pao, Quasisolutions and global attractor of reaction-diffusion systems, Nonlinear Anal., 26 (1996), 1889-1903. doi: 10.1016/0362-546X(95)00058-4.

[35]

G. A. PolisC. A. Myers and R. D. Holt, The ecology and evolution of intraguild predation: Potential competitors that eat each other, Annual Review of Ecology and Systematics, 20 (1989), 297-330. doi: 10.1146/annurev.es.20.110189.001501.

[36]

G. A. Polis and R. D. Holt, Intraguild predation: the dynamics of complex trophic interactions, Trends in Ecology and Evolution, 7 (1992), 151-154. doi: 10.1016/0169-5347(92)90208-S.

[37]

D. Ryan and R. S. Cantrell, Aoidance behavior in intraguild predation communities: A cross-diffusion model, Discrete Contin. Dyn. Syst. A, 35 (2015), 1641-1663.

[38]

T. Schellekens and T. V. Kooten, Coexistence of two stage-structured intraguild predators, J. Theor. Biol., 308 (2012), 36-44. doi: 10.1016/j.jtbi.2012.05.017.

[39]

H. ShuX. HuL. Wang and J. Watmough, Delay induced stability switch, multitype bistability and chaos in an intraguild predation model, J. Math. Biol., 71 (2015), 1269-1298. doi: 10.1007/s00285-015-0857-4.

[40]

J. Smoller, Shock Waves and Reaction-Diffusion Equations Second edition, in Grundkehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), 258, Spring-Verlag New York, 1994.

[41]

K. Tanabe and T. Namba, Omnivory creates chaos in simple food webmodels, Ecology, 86 (2005), 3411-3414.

[42]

P. Urbania and R. R. Jiliberto, Adaptive prey behavior and the dynamics of intraguild predation systems, Ecological Modelling, 221 (2010), 2628-2633. doi: 10.1016/j.ecolmodel.2010.08.009.

[43]

A. Verdy and P. Amarasekare, Alternative stable states in communities with intraguild predation, J. Theor. Biol., 262 (2010), 116-128. doi: 10.1016/j.jtbi.2009.09.011.

[44]

A. W. VisserP. Mariani and S. Pigolotti, Adaptive behaviour, tri-trophic foodweb stability and damping of chaos, Journal of the Royal Society Interface, 9 (2012), 1373-1380. doi: 10.1098/rsif.2011.0686.

[45]

M. YamaguchiY. Takeuchi and W. Ma, Dynamical properties of a stage structured three-species model with intra-guild predation, Journal of Computational and Applied Mathematics, 201 (2007), 327-338. doi: 10.1016/j.cam.2005.12.033.

[46]

G. ZhangW. Wang and X. Wang, Coexistence states for a diffusive one-prey and two-predators model with B-D functional response, J. Math. Anal. Appl., 387 (2012), 931-948. doi: 10.1016/j.jmaa.2011.09.049.

[47]

J. Zhou and C. Mu, Positive solutions for a three-trophic food chain model with diffusion and Beddington-Deangelis functional response, Nonlinear Anal. Real World Appl, 12 (2011), 902-917. doi: 10.1016/j.nonrwa.2010.08.015.

Figure 1.  Coexistence induced by $r_{1}$ with fixed parameter values $a_{1} = 5$, $a_{3} = 3$, $a_{3} = 3$, $m_{1} = 2.5$, $m_{2} = 1.5$, $m_{3} = 1$, $b_{1} = b_{2} = b_{3} = 0.2$, $r_{2} = 0.3$, $r_{3} = 0.2$, $e_{1} = e_{2} = e_{3} = 1$
Figure 2.  Coexistence induced by $r_{2}$ with fixed parameter values $a_{1} = 5$, $a_{3} = 3$, $a_{3} = 2$, $m_{1} = 2.5$, $m_{2} = 1.5$, $m_{3} = 1$, $b_{1} = b_{2} = b_{3} = 0.2$, $r_{1} = 2$, $r_{3} = 0.2$, $e_{1} = e_{2} = e_{3} = 1$
Figure 3.  Coexistence and extinction induced by $r_{3}$ with fixed parameter values $a_{1} = 5$, $a_{3} = 3$, $a_{3} = 2$, $m_{1} = 2.5$, $m_{2} = 1.5$, $m_{3} = 1$, $b_{1} = b_{2} = b_{3} = 0.2$, $r_{1} = 2$, $r_{2} = 0.3$, $e_{1} = e_{2} = e_{3} = 1$
Figure 4.  Coexistence induced by interference of IGpredators ($e_{2}$) with fixed parameter values $a_{1} = 5$, $a_{3} = 3$, $a_{3} = 2$, $m_{1} = 2.5$, $m_{2} = 1.5$, $m_{3} = 1$, $b_{1} = b_{2} = b_{3} = 0.2$, $r_{1} = 2$, $r_{2} = 0.3$, $r_{3} = 0.2, e_{1} = e_{3} = 1$
Figure 5.  Coexistence induced by interference among IGpredators ($e_{3}$) with fixed parameter values $a_{1} = 5$, $a_{3} = 3$, $a_{3} = 2$, $m_{1} = 2.5$, $m_{2} = 1.5$, $m_{3} = 1$, $b_{1} = b_{2} = b_{3} = 0.2$, $r_{1} = 2$, $r_{2} = 0.3$, $r_{3} = 0.2$, $e_{1} = e_{2} = 1$
Figure 6.  Extinction of IGprey driven by interference among IGprey ($e_{1}$) with fixed parameter values $a_{1} = 5$, $a_{3} = 3$, $a_{3} = 2$, $m_{1} = 2.5$, $m_{2} = 1.5$, $m_{3} = 1$, $b_{1} = b_{2} = b_{3} = 0.2$, $r_{1} = 2$, $r_{2} = 0.3$, $r_{3} = 0.2$, $e_{2} = e_{3} = 1$
Figure 7.  Extinction of IGpredator driven by $e_{2}$ and $e_{3}$ with fixed parameter values $a_{1} = 5$, $a_{3} = 3$, $a_{3} = 2$, $m_{1} = 2.5$, $m_{2} = 1.5$, $m_{3} = 1$, $b_{1} = b_{2} = b_{3} = 0.2$, $r_{1} = 2$, $r_{2} = 0.3$, $r_{3} = -0.5$, $e_{1} = 1$
Figure 8.  Coexistence induced by IGP ($a_{2}$ and $m_{2}$) with fixed parameter values $a_{1} = 5$, $a_{3} = 0.5$, $m_{1} = 2.5$, $m_{3} = 0.25$, $b_{1} = b_{2} = b_{3} = 0.2$, $r_{1} = 2$, $r_{2} = 0.1$, $r_{3} = -0.15$, $e_{1} = e_{2} = e_{3} = 10$
Figure 9.  Extinction of IGprey driven by IGP ($a_{2}$ and $m_{2}$) with fixed parameter values $a_{1} = 1$, $a_{3} = 0.5$, $m_{1} = 0.5$, $m_{3} = 0.25$, $b_{1} = b_{2} = b_{3} = 0.2$, $r_{1} = 2$, $r_{2} = 0.1$, $r_{3} = -0.15$, $e_{1} = e_{2} = e_{3} = 1$
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