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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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.   Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.   Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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