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2018, 15(3): 595-627. doi: 10.3934/mbe.2018027

## Delay induced spatiotemporal patterns in a diffusive intraguild predation model with Beddington-DeAngelis functional response

 1 School of Science, Zhejiang University of Science & Technology, Hangzhou, 310023, China 2 School of Mathematics and Statistics, Central South University, Changsha, 410083, China 3 Department of Mathematics and Statistics, University of New Brunswick, Fredericton, E3B 5A3, Canada

* Corresponding author: bxdai@csu.edu.cn

Received  February 16, 2017 Revised  August 3, 2017 Published  December 2017

A diffusive intraguild predation model with delay and Beddington-DeAngelis functional response is considered. Dynamics including stability and Hopf bifurcation near the spatially homogeneous steady states are investigated in detail. Further, it is numerically demonstrated that delay can trigger the emergence of irregular spatial patterns including chaos. The impacts of diffusion and functional response on the model's dynamics are also numerically explored.

Citation: Renji Han, Binxiang Dai, Lin Wang. Delay induced spatiotemporal patterns in a diffusive intraguild predation model with Beddington-DeAngelis functional response. Mathematical Biosciences & Engineering, 2018, 15 (3) : 595-627. doi: 10.3934/mbe.2018027
##### References:
 [1] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Spring-Verlag, New York, 2003. [2] G. A. Polis, C. A. Myers, R. D. Holt, The ecology and evolution of intraguild predation: Potential competitors that each other, Ann. Rev. Ecol. Sys., 20 (1989), 297-330. doi: 10.1146/annurev.es.20.110189.001501. [3] M. H. Posey, A. H. Hines, Complex predator-prey interactions within an estuarine benthic community, Ecol., 72 (1991), 2155-2169. doi: 10.2307/1941567. [4] G. A. Polis, R. D. Holt, Intraguild predation: The dynamics of complex trophic interactions, Trends Ecol. Evol., 7 (1992), 151-154. [5] R. D. Holt, G. A. Polis, A theoretical framework for intraguild predation, Am. Nat., 149 (1997), 745-764. doi: 10.1086/286018. [6] M. Arim, P. A. Marquet, Intraguild predation: A widespread interaction related to species biology, Ecol. Let., 7 (2004), 557-564. doi: 10.1111/j.1461-0248.2004.00613.x. [7] P. Amarasekare, Trade-offs, temporal, variation, and species coexistence in communities with intraguild predation, Ecol., 88 (2007), 2720-2728. doi: 10.1890/06-1515.1. [8] R. Hall, Intraguild predation in the presence of a shared natural enemy, Ecol., 92 (2011), 352-361. doi: 10.1890/09-2314.1. [9] Y. S. Wang, D. L. DeAngelis, Stability of an intraguild predation system with mutual predation, Commun. Nonlinear Sci. Numer. Simulat., 33 (2016), 141-159. doi: 10.1016/j.cnsns.2015.09.004. [10] I. Velazquez, D. Kaplan, J. X. Velasco-Hernandez, S. A. Navarrete, Multistability in an open recruitment food web model, Appl. Math. Comp., 163 (2005), 275-294. doi: 10.1016/j.amc.2004.02.005. [11] S. B. Hsu, S. Ruan, 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. [12] P. A. Abrams, S. R. Fung, Prey persistence and abundance in systems with intraguild predation and type-2 functional response, J. Theor. Biol., 264 (2010), 1033-1042. doi: 10.1016/j.jtbi.2010.02.045. [13] A. Verdy, P. Amarasekare, Alternative stable states in communities with intraguild predatiion, J. Theor. Biol., 262 (2010), 116-128. doi: 10.1016/j.jtbi.2009.09.011. [14] M. Freeze, Y. Chang, W. Feng, Analysis of dynamics in a complex food chain with ratio-dependent functional response, J. Appl. Anal. Comput., 4 (2014), 69-87. [15] Y. Kang, 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. [16] H. I. Freedman, V. S. H. Rao, Stability criteria for a system involving two time delays, SIAM J. Appl. Math., 46 (1986), 552-560. doi: 10.1137/0146037. [17] G. S. K. Wolkowicz, H. X. Xia, Global asymptotic behavior of chemostat model with discrete delays, SIAM J. Appl. Math., 57 (1997), 1019-1043. doi: 10.1137/S0036139995287314. [18] Y. L. Song, M. A. Han, J. J. Wei, Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays, Physica D, 200 (2005), 185-204. doi: 10.1016/j.physd.2004.10.010. [19] S. Ruan, On nonlinear dynamics of predator-prey models with discrete delay, Math. Mod. Nat. Phen., 4 (2009), 140-188. doi: 10.1051/mmnp/20094207. [20] X. Y. Meng, H. F. Huo, X. B. Zhao, H. Xiang, Stability and Hopf bifurcation in a three-species system with feedback delays, Nonlinear Dyn., 64 (2011), 349-364. doi: 10.1007/s11071-010-9866-4. [21] M. Y. Li, H. Shu, Multiple stable periodic oscillations in a mathematical model of CTL-response to HTLV-I infection, Bull. Math. Biol., 73 (2011), 1774-1793. doi: 10.1007/s11538-010-9591-7. [22] H. Shu, L. Wang, J. Watmough, Sustained and transient oscillations and chaos induced by delayed antiviral inmune response in an immunosuppressive infective model, J. Math. Biol., 68 (2014), 477-503. doi: 10.1007/s00285-012-0639-1. [23] M. Yamaguchi, Y. Takeuchi, W. Ma, Dynamical properties of a stage structured three-species model with intra-guild predation, J. Comput. Appl. Math., 201 (2007), 327-338. doi: 10.1016/j.cam.2005.12.033. [24] H. Shu, X. Hu, L. Wang, J. Watmough, Delayed 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. [25] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern perspectives, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6. [26] T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463. doi: 10.1006/jmaa.2000.7182. [27] C. V. Pao, Systems of parabolic equations with continuous and discrete delays, J. Math. Anal. Appl., 205 (1997), 157-185. doi: 10.1006/jmaa.1996.5177. [28] C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Anal., 48 (2002), 349-362. doi: 10.1016/S0362-546X(00)00189-9. [29] J. Wang, J. P. Shi, J. J. Wei, Dyanmics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Diff. Equat., 251 (2011), 1276-1304. doi: 10.1016/j.jde.2011.03.004. [30] C. Tian, Delay-driven spatial patterns in a plankton allelopathic system, Chaos, 22(2012), 013129, 7 pp. doi: 10.1063/1.3692963. [31] C. Tian, L. Zhang, Hopf bifurcation analysis in a diffusive food-chain model with time delay, Comput. Math. Appl., 66 (2013), 2139-2153. doi: 10.1016/j.camwa.2013.09.002. [32] W. Zuo, J. Wei, Global stability and Hopf bifurcations of a Beddington-DeAngelis type predator-prey system with diffusion and delay, Appl. Math. Comput., 223 (2013), 423-435. doi: 10.1016/j.amc.2013.08.029. [33] J. Zhao, J. Wei, Dynamics in a diffusive plankton system with delay and toxic substances effect, Nonlinear Anal., 22 (2015), 66-83. doi: 10.1016/j.nonrwa.2014.07.010. [34] L. Zhu, H. Zhao, X. M. Wang, Bifurcation analysis of a delay reaction-diffusion malware propagation model with feedback control, Commun. Nonlinear Sci. Numer. Simulat., 22 (2015), 747-768. doi: 10.1016/j.cnsns.2014.08.027. [35] Y. Li, M. X. Wang, Hopf bifurcation and global stability of a delayed predator-prey model with prey harvesting, Comput. Math. Appl., 69 (2015), 398-410. doi: 10.1016/j.camwa.2015.01.003. [36] H. Y. Zhao, X. Zhang, X. Huang, Hopf bifurcation and spatial patterns of a delayed biological economic system with diffusion, Appl. Math. Comput., 266 (2015), 462-480. doi: 10.1016/j.amc.2015.05.089. [37] Q. X. Ye, Z. Y. Li, M. X. Wang and Y. P. Wu, Introduction to Reaction-diffusion Equations (Second Edition), Science Press, Bei Jing, 2011. [38] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin/New York, 1981. [39] S. Ruan, J. Wei, On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testoterone secretion, Math. Med. Biol., 18 (2001), 41-52. [40] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [41] B. Hassard, N. Kazarinoff and Y. Wan, Theory and Applications of Hopf bifurcation, Cambridge University Press, Cambridge, 1981. [42] J. Y. Wakano, C. Hauert, Pattern formation and chaos in spatial ecological public goods games, J. Theor. Biol., 268 (2011), 30-38. doi: 10.1016/j.jtbi.2010.09.036. [43] M. Banerjee, S. Ghoral and N. Mukherjee, Approximated spiral and target patterns in Bazykin's prey-predator model: Multiscale perturbation analysis, Int. J. Bifurcat. Chaos, 27 (2017), 1750038, 14 pp. doi: 10.1142/S0218127417500389. [44] H. Malchow, S. V. Petrovskii and E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, Simulations, Chapman & Hall / CRC Press, 2008. [45] Q. Ouyang, Pattern Formation in Reaction-Diffusion Systems Shanghai Scientific and Technological Education Publishing House, SHANGHAI, 2000.

show all references

##### References:
 [1] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Spring-Verlag, New York, 2003. [2] G. A. Polis, C. A. Myers, R. D. Holt, The ecology and evolution of intraguild predation: Potential competitors that each other, Ann. Rev. Ecol. Sys., 20 (1989), 297-330. doi: 10.1146/annurev.es.20.110189.001501. [3] M. H. Posey, A. H. Hines, Complex predator-prey interactions within an estuarine benthic community, Ecol., 72 (1991), 2155-2169. doi: 10.2307/1941567. [4] G. A. Polis, R. D. Holt, Intraguild predation: The dynamics of complex trophic interactions, Trends Ecol. Evol., 7 (1992), 151-154. [5] R. D. Holt, G. A. Polis, A theoretical framework for intraguild predation, Am. Nat., 149 (1997), 745-764. doi: 10.1086/286018. [6] M. Arim, P. A. Marquet, Intraguild predation: A widespread interaction related to species biology, Ecol. Let., 7 (2004), 557-564. doi: 10.1111/j.1461-0248.2004.00613.x. [7] P. Amarasekare, Trade-offs, temporal, variation, and species coexistence in communities with intraguild predation, Ecol., 88 (2007), 2720-2728. doi: 10.1890/06-1515.1. [8] R. Hall, Intraguild predation in the presence of a shared natural enemy, Ecol., 92 (2011), 352-361. doi: 10.1890/09-2314.1. [9] Y. S. Wang, D. L. DeAngelis, Stability of an intraguild predation system with mutual predation, Commun. Nonlinear Sci. Numer. Simulat., 33 (2016), 141-159. doi: 10.1016/j.cnsns.2015.09.004. [10] I. Velazquez, D. Kaplan, J. X. Velasco-Hernandez, S. A. Navarrete, Multistability in an open recruitment food web model, Appl. Math. Comp., 163 (2005), 275-294. doi: 10.1016/j.amc.2004.02.005. [11] S. B. Hsu, S. Ruan, 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. [12] P. A. Abrams, S. R. Fung, Prey persistence and abundance in systems with intraguild predation and type-2 functional response, J. Theor. Biol., 264 (2010), 1033-1042. doi: 10.1016/j.jtbi.2010.02.045. [13] A. Verdy, P. Amarasekare, Alternative stable states in communities with intraguild predatiion, J. Theor. Biol., 262 (2010), 116-128. doi: 10.1016/j.jtbi.2009.09.011. [14] M. Freeze, Y. Chang, W. Feng, Analysis of dynamics in a complex food chain with ratio-dependent functional response, J. Appl. Anal. Comput., 4 (2014), 69-87. [15] Y. Kang, 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. [16] H. I. Freedman, V. S. H. Rao, Stability criteria for a system involving two time delays, SIAM J. Appl. Math., 46 (1986), 552-560. doi: 10.1137/0146037. [17] G. S. K. Wolkowicz, H. X. Xia, Global asymptotic behavior of chemostat model with discrete delays, SIAM J. Appl. Math., 57 (1997), 1019-1043. doi: 10.1137/S0036139995287314. [18] Y. L. Song, M. A. Han, J. J. Wei, Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays, Physica D, 200 (2005), 185-204. doi: 10.1016/j.physd.2004.10.010. [19] S. Ruan, On nonlinear dynamics of predator-prey models with discrete delay, Math. Mod. Nat. Phen., 4 (2009), 140-188. doi: 10.1051/mmnp/20094207. [20] X. Y. Meng, H. F. Huo, X. B. Zhao, H. Xiang, Stability and Hopf bifurcation in a three-species system with feedback delays, Nonlinear Dyn., 64 (2011), 349-364. doi: 10.1007/s11071-010-9866-4. [21] M. Y. Li, H. Shu, Multiple stable periodic oscillations in a mathematical model of CTL-response to HTLV-I infection, Bull. Math. Biol., 73 (2011), 1774-1793. doi: 10.1007/s11538-010-9591-7. [22] H. Shu, L. Wang, J. Watmough, Sustained and transient oscillations and chaos induced by delayed antiviral inmune response in an immunosuppressive infective model, J. Math. Biol., 68 (2014), 477-503. doi: 10.1007/s00285-012-0639-1. [23] M. Yamaguchi, Y. Takeuchi, W. Ma, Dynamical properties of a stage structured three-species model with intra-guild predation, J. Comput. Appl. Math., 201 (2007), 327-338. doi: 10.1016/j.cam.2005.12.033. [24] H. Shu, X. Hu, L. Wang, J. Watmough, Delayed 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. [25] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern perspectives, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6. [26] T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463. doi: 10.1006/jmaa.2000.7182. [27] C. V. Pao, Systems of parabolic equations with continuous and discrete delays, J. Math. Anal. Appl., 205 (1997), 157-185. doi: 10.1006/jmaa.1996.5177. [28] C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Anal., 48 (2002), 349-362. doi: 10.1016/S0362-546X(00)00189-9. [29] J. Wang, J. P. Shi, J. J. Wei, Dyanmics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Diff. Equat., 251 (2011), 1276-1304. doi: 10.1016/j.jde.2011.03.004. [30] C. Tian, Delay-driven spatial patterns in a plankton allelopathic system, Chaos, 22(2012), 013129, 7 pp. doi: 10.1063/1.3692963. [31] C. Tian, L. Zhang, Hopf bifurcation analysis in a diffusive food-chain model with time delay, Comput. Math. Appl., 66 (2013), 2139-2153. doi: 10.1016/j.camwa.2013.09.002. [32] W. Zuo, J. Wei, Global stability and Hopf bifurcations of a Beddington-DeAngelis type predator-prey system with diffusion and delay, Appl. Math. Comput., 223 (2013), 423-435. doi: 10.1016/j.amc.2013.08.029. [33] J. Zhao, J. Wei, Dynamics in a diffusive plankton system with delay and toxic substances effect, Nonlinear Anal., 22 (2015), 66-83. doi: 10.1016/j.nonrwa.2014.07.010. [34] L. Zhu, H. Zhao, X. M. Wang, Bifurcation analysis of a delay reaction-diffusion malware propagation model with feedback control, Commun. Nonlinear Sci. Numer. Simulat., 22 (2015), 747-768. doi: 10.1016/j.cnsns.2014.08.027. [35] Y. Li, M. X. Wang, Hopf bifurcation and global stability of a delayed predator-prey model with prey harvesting, Comput. Math. Appl., 69 (2015), 398-410. doi: 10.1016/j.camwa.2015.01.003. [36] H. Y. Zhao, X. Zhang, X. Huang, Hopf bifurcation and spatial patterns of a delayed biological economic system with diffusion, Appl. Math. Comput., 266 (2015), 462-480. doi: 10.1016/j.amc.2015.05.089. [37] Q. X. Ye, Z. Y. Li, M. X. Wang and Y. P. Wu, Introduction to Reaction-diffusion Equations (Second Edition), Science Press, Bei Jing, 2011. [38] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin/New York, 1981. [39] S. Ruan, J. Wei, On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testoterone secretion, Math. Med. Biol., 18 (2001), 41-52. [40] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [41] B. Hassard, N. Kazarinoff and Y. Wan, Theory and Applications of Hopf bifurcation, Cambridge University Press, Cambridge, 1981. [42] J. Y. Wakano, C. Hauert, Pattern formation and chaos in spatial ecological public goods games, J. Theor. Biol., 268 (2011), 30-38. doi: 10.1016/j.jtbi.2010.09.036. [43] M. Banerjee, S. Ghoral and N. Mukherjee, Approximated spiral and target patterns in Bazykin's prey-predator model: Multiscale perturbation analysis, Int. J. Bifurcat. Chaos, 27 (2017), 1750038, 14 pp. doi: 10.1142/S0218127417500389. [44] H. Malchow, S. V. Petrovskii and E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, Simulations, Chapman & Hall / CRC Press, 2008. [45] Q. Ouyang, Pattern Formation in Reaction-Diffusion Systems Shanghai Scientific and Technological Education Publishing House, SHANGHAI, 2000.
Numerical solutions of (4) with $\tau = 0.7<\tau^\ast\approx0.7895$ (only the $u_1$ component is plotted here): the positive spatially homogeneous steady state is locally stable.
Numerical solutions of (4) with $\tau = 1.2>\tau^\ast\approx0.7895$: a periodic solution bifurcates from the positive spatially homogeneous steady state $E^\ast$.
Numerical solutions of the temporal model (left) and numerical solutions of the spatiotemporal model (right) with $\tau = 1$, $(P_2)$ and $(IC_2).$ Here, for the spatiotemporal model (4), average population density for each species is plotted.
Numerical solutions of the temporal model (left) and numerical solutions of the spatiotemporal model (right) with $\tau = 1.5$, $(P_2)$ and $(IC_2').$ Here, periodic oscillations are observed for the temporal model and chaotic behavior is observed for the spatiotemporal model.
Snapshots of contour maps of the basal resource $u_1$ for the temporal model (left) and spatiotemporal model (right) at $t = 2000$ with $\tau = 1.5$, $(P_2)$ and $(IC_2^\prime).$
Snapshots of contour maps of the time evolution of the specie $u_1$ at $t = 200$, $500$, $1000$, $1200$, $1500$, $2500$ with $\tau = 1.5$ under $(P_2)$ and $(IC_2^\prime)$.
Snapshots of contour maps of the time evolution of the basal resource $u_1$ with different values of $\tau$ at time $t = 1500$ under $(P_2)$ and $(IC_2^\prime)$. $(\mathrm{ⅰ})\,\tau = 0.86; (\mathrm{ⅱ})\,\tau = 1; (\mathrm{ⅲ})\,\tau = 1.2; (\mathrm{ⅳ})\,\tau = 1.4; (\mathrm{ⅴ})\,\tau = 1.6; (\mathrm{ⅵ})\,\tau = 1.9.$
Snapshots of contour maps of the basal resource $u_1$ at time $t = 1500$ with different diffusion coefficients, $\tau = 1.5$, under $(P_2)$ and $(IC_2^\prime)$.
Snapshots of contour maps of the time evolution of the basal resource $u_1$ with different values of $b$ and parameter values $\alpha = 0.7, \beta = 0.9, \beta_1 = 1.95, \beta_2 = 1.85, \gamma_1 = 0.2, \gamma_2 = 0.8, c = 5$ at times $t = 1500$ and $\tau = 1.5$ under$(IC_2^\prime)$.
10. Snapshots of contour maps of the time evolution of the basal resource $u_1$ with different values of $c$ and parameter values $\alpha = 0.7, \beta = 0.9, \beta_1 = 1.95, \beta_2 = 1.85, \gamma_1 = 0.2, \gamma_2 = 0.8, b = 0.25$ at times $t = 1500$ and $\tau = 1.5$ under$(IC_2^\prime)$.
Parameters definitions in model (3) and their units, where [resource] indicates basal resource density, [IG prey] indicates IG prey density, and [IG predator] indicates IG predator density
 Symbol Parameter Definition Units $r$ Basal resource intrinsic growth rate [time]$^{-1}$ $K$ Basal resource carrying capacity [Basal resource density] $c_1$ Predation rate of IG prey on resource [IG prey]$^{-1}$ [time]$^{-1}$ $c_2$ Predation rate of IG predator on resource [IG predator]$^{-1}$[time]$^{-1}$ $c_3$ Predation rate of IG predaotr on IG prey [IG preys][IG predator]$^{-1}$ [time]$^{-1}$ $e_1$ Conversion rate from resource to IG prey [IG preys][resource]$^{-1}$ $e_2$ Conversion rate from resource to IG predator [IG predators][resource]$^{-1}$ $e_3$ Conversion rate from IG prey to IG predator [IG predators][IG prey]$^{-1}$ $a_1$ [Half saturation constant]$^{-1}$ [resource]$^{-1}$ $a_2$ [Half saturation constant]$^{-1}$ [IG predator]$^{-1}$ $m_1$ Mortality rate of IG prey [time]$^{-1}$ $m_2$ Mortality rate of IG predator [time]$^{-1}$ ${\widetilde d}_1$ Diffusion coefficient of resource [length]$^2$[time]$^{-1}$ ${\widetilde d}_2$ Diffusion coefficient of IG prey [length]$^2$[time]$^{-1}$ ${\widetilde d}_3$ Diffusion coefficient of IG predatior [length]$^2$[time]$^{-1}$ $L$ The size of spatial domain $\Omega$ [length]
 Symbol Parameter Definition Units $r$ Basal resource intrinsic growth rate [time]$^{-1}$ $K$ Basal resource carrying capacity [Basal resource density] $c_1$ Predation rate of IG prey on resource [IG prey]$^{-1}$ [time]$^{-1}$ $c_2$ Predation rate of IG predator on resource [IG predator]$^{-1}$[time]$^{-1}$ $c_3$ Predation rate of IG predaotr on IG prey [IG preys][IG predator]$^{-1}$ [time]$^{-1}$ $e_1$ Conversion rate from resource to IG prey [IG preys][resource]$^{-1}$ $e_2$ Conversion rate from resource to IG predator [IG predators][resource]$^{-1}$ $e_3$ Conversion rate from IG prey to IG predator [IG predators][IG prey]$^{-1}$ $a_1$ [Half saturation constant]$^{-1}$ [resource]$^{-1}$ $a_2$ [Half saturation constant]$^{-1}$ [IG predator]$^{-1}$ $m_1$ Mortality rate of IG prey [time]$^{-1}$ $m_2$ Mortality rate of IG predator [time]$^{-1}$ ${\widetilde d}_1$ Diffusion coefficient of resource [length]$^2$[time]$^{-1}$ ${\widetilde d}_2$ Diffusion coefficient of IG prey [length]$^2$[time]$^{-1}$ ${\widetilde d}_3$ Diffusion coefficient of IG predatior [length]$^2$[time]$^{-1}$ $L$ The size of spatial domain $\Omega$ [length]
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