August  2020, 13(8): 2109-2120. doi: 10.3934/dcdss.2020180

Prey-predator model with nonlocal and global consumption in the prey dynamics

1. 

Department of Mathematics & Statistics, Indian Institute of Technology Kanpur, Kanpur - 208016, India

2. 

Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France

3. 

INRIA, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 Bd. du 11 Novembre 1918, 69200 Villeurbanne Cedex, France

4. 

Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

* Corresponding author: Vitaly Volpert

Received  January 2019 Published  August 2020 Early access  November 2019

Fund Project: The third author is supported by "RUDN University Program 5-100"

A prey-predator model with a nonlocal or global consumption of resources by prey is studied. Linear stability analysis about the homogeneous in space stationary solution is carried out to determine the conditions of the bifurcation of stationary and moving pulses in the case of global consumption. Their existence is confirmed in numerical simulations. Periodic travelling waves and multiple pulses are observed for the nonlocal consumption.

Citation: Malay Banerjee, Nayana Mukherjee, Vitaly Volpert. Prey-predator model with nonlocal and global consumption in the prey dynamics. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2109-2120. doi: 10.3934/dcdss.2020180
References:
[1]

A. ApreuteseiA. Ducrot and V. Volpert, Competition of species with intra-specific competition, Math. Model. Nat. Phenom., 3 (2008), 1-27.  doi: 10.1051/mmnp:2008068.

[2]

N. ApreuteseiA. Ducrot and V. Volpert, Travelling waves for integro-differential equations in population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 541-561.  doi: 10.3934/dcdsb.2009.11.541.

[3]

N. ApreuteseiN. BessonovV. Volpert and V. Vougalter, Sptaial structures and generalized travelling waves for an integro-differential equation, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 537-557.  doi: 10.3934/dcdsb.2010.13.537.

[4]

O. Aydogmus, Patterns and transitions to instability in an intraspecific competition model with nonlocal diffusion and interaction, Math. Model. Nat. Phenom., 10 (2015), 17-19.  doi: 10.1051/mmnp/201510603.

[5]

M. Banerjee and S. Banerjee, Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model, Math. Biosci., 236 (2012), 64-76.  doi: 10.1016/j.mbs.2011.12.005.

[6]

M. BanerjeeN. Mukherjee and V. Volpert, Prey-predator model with a nonlocal bistable dynamics of prey, Mathematics, 6 (2018), 1-13.  doi: 10.3390/math6030041.

[7]

M. Banerjee and S. Petrovskii, Self-organized spatial patterns and chaos in a ratio-dependent predator-prey system, Theor. Ecol., 4 (2011), 37-53. 

[8]

M. Banerjee and V. Volpert, Prey-predator model with a nonlocal consumption of prey, Chaos, 26 (2016), 12pp. doi: 10.1063/1.4961248.

[9]

M. Banerjee and V. Volpert, Spatio-temporal pattern formation in Rosenzweig-McArthur model: Effect of nonlocal interactions, Ecol. Complex., 30 (2017), 2-10. 

[10]

M. BanerjeeV. Vougalter and V. Volpert, Doubly nonlocal reaction–diffusion equations and the emergence of species, Appl. Math. Model., 42 (2017), 591-599.  doi: 10.1016/j.apm.2016.10.041.

[11]

M. BaurmannW. Ebenhoh and U. Feudel, Turing instabilities and pattern formation in a benthic nutrient-microorganism system, Math. Biosci. Eng., 1 (2004), 111-130.  doi: 10.3934/mbe.2004.1.111.

[12]

M. BaurmannT. Gross and U. Feudel, Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J. Theoret. Biol., 245 (2007), 220-229.  doi: 10.1016/j.jtbi.2006.09.036.

[13]

A. Bayliss and V. A. Volpert, Patterns for competing populations with species specific nonlocal coupling, Math. Model. Nat. Phenom., 10 (2015), 30-47.  doi: 10.1051/mmnp/201510604.

[14]

N. BessonovN. Reinberg and V. Volpert, Mathematics of Darwin's diagram, Math. Model. Nat. Phenom., 9 (2014), 5-25.  doi: 10.1051/mmnp/20149302.

[15]

S. Fasani and S. Rinaldi, Factors promoting or inhibiting Turing instability in spatially extended prey-predator systems, Ecol. Model., 222 (2011), 3449-3452.  doi: 10.1016/j.ecolmodel.2011.07.002.

[16]

T. GalochkinaM. Marion and V. Volpert, Initiation of reaction-diffusion waves of blood coagulation, Phys. D, 376-377 (2018), 160-170.  doi: 10.1016/j.physd.2017.11.006.

[17]

G. F. Gause, The Struggle for Existence, Williams and Wilkins, Baltimore, 1934. doi: 10.5962/bhl.title.4489.

[18]

S. GenieysN. Bessonov and V. Volpert, Mathematical model of evolutionary branching, Math. Comput. Modelling, 49 (2009), 2109-2115.  doi: 10.1016/j.mcm.2008.07.018.

[19]

S. GenieysV. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Model. Nat. Phenom., 1 (2006), 65-82.  doi: 10.1051/mmnp:2006004.

[20]

S. GenieysV. Volpert and P. Auger, Adaptive dynamics: Modelling Darwin's divergence principle, Comp. Ren. Biol., 329 (2006), 876-879.  doi: 10.1016/j.crvi.2006.08.006.

[21]

J. D. Murray, Mathematical Biology. Ⅱ: Spatial Models And Biomedical Applications, Interdisciplinary Applied Mathematics, 19, Springer-Verlag, New York, 2003. doi: 10.1007/b98869.

[22]

S. PalS. Ghorai and M. Banerjee, Analysis of a prey-predator model with non-local interaction in the prey population, Bull. Math. Biol., 80 (2018), 906-925.  doi: 10.1007/s11538-018-0410-x.

[23]

S. V. Petrovskii and H. Malchow, A minimal model of pattern formation in a prey-predator system, Math. Comput. Modelling, 29 (1999), 49-63.  doi: 10.1016/S0895-7177(99)00070-9.

[24]

J. A. SherrattB. T. Eagan and M. A. Lewis, Oscillations and chaos behind predator-prey invasion: Mathematical artifact or ecological reality?, Phil. Trans. R. Soc. Lond. B, 352 (1997), 21-38.  doi: 10.1098/rstb.1997.0003.

[25]

V. Volpert, Branching and aggregation in self-reproducing systems, in MMCS, Mathematical Modelling of Complex Systems, ESAIM Proc. Surveys, 47, EDP Sci., Les Ulis, 2014,116–129. doi: 10.1051/proc/201447007.

[26]

V. Volpert, Elliptic Partial Differential Equations, Monographs in Mathematics, 104, Birkhäuser/Springer Basel AG, Basel, 2014. doi: 10.1007/978-3-0348-0813-2.

[27]

V. Volpert, Pulses and waves for a bistable nonlocal reaction-diffusion equation, Appl. Math. Lett., 44 (2015), 21-25.  doi: 10.1016/j.aml.2014.12.011.

show all references

References:
[1]

A. ApreuteseiA. Ducrot and V. Volpert, Competition of species with intra-specific competition, Math. Model. Nat. Phenom., 3 (2008), 1-27.  doi: 10.1051/mmnp:2008068.

[2]

N. ApreuteseiA. Ducrot and V. Volpert, Travelling waves for integro-differential equations in population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 541-561.  doi: 10.3934/dcdsb.2009.11.541.

[3]

N. ApreuteseiN. BessonovV. Volpert and V. Vougalter, Sptaial structures and generalized travelling waves for an integro-differential equation, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 537-557.  doi: 10.3934/dcdsb.2010.13.537.

[4]

O. Aydogmus, Patterns and transitions to instability in an intraspecific competition model with nonlocal diffusion and interaction, Math. Model. Nat. Phenom., 10 (2015), 17-19.  doi: 10.1051/mmnp/201510603.

[5]

M. Banerjee and S. Banerjee, Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model, Math. Biosci., 236 (2012), 64-76.  doi: 10.1016/j.mbs.2011.12.005.

[6]

M. BanerjeeN. Mukherjee and V. Volpert, Prey-predator model with a nonlocal bistable dynamics of prey, Mathematics, 6 (2018), 1-13.  doi: 10.3390/math6030041.

[7]

M. Banerjee and S. Petrovskii, Self-organized spatial patterns and chaos in a ratio-dependent predator-prey system, Theor. Ecol., 4 (2011), 37-53. 

[8]

M. Banerjee and V. Volpert, Prey-predator model with a nonlocal consumption of prey, Chaos, 26 (2016), 12pp. doi: 10.1063/1.4961248.

[9]

M. Banerjee and V. Volpert, Spatio-temporal pattern formation in Rosenzweig-McArthur model: Effect of nonlocal interactions, Ecol. Complex., 30 (2017), 2-10. 

[10]

M. BanerjeeV. Vougalter and V. Volpert, Doubly nonlocal reaction–diffusion equations and the emergence of species, Appl. Math. Model., 42 (2017), 591-599.  doi: 10.1016/j.apm.2016.10.041.

[11]

M. BaurmannW. Ebenhoh and U. Feudel, Turing instabilities and pattern formation in a benthic nutrient-microorganism system, Math. Biosci. Eng., 1 (2004), 111-130.  doi: 10.3934/mbe.2004.1.111.

[12]

M. BaurmannT. Gross and U. Feudel, Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J. Theoret. Biol., 245 (2007), 220-229.  doi: 10.1016/j.jtbi.2006.09.036.

[13]

A. Bayliss and V. A. Volpert, Patterns for competing populations with species specific nonlocal coupling, Math. Model. Nat. Phenom., 10 (2015), 30-47.  doi: 10.1051/mmnp/201510604.

[14]

N. BessonovN. Reinberg and V. Volpert, Mathematics of Darwin's diagram, Math. Model. Nat. Phenom., 9 (2014), 5-25.  doi: 10.1051/mmnp/20149302.

[15]

S. Fasani and S. Rinaldi, Factors promoting or inhibiting Turing instability in spatially extended prey-predator systems, Ecol. Model., 222 (2011), 3449-3452.  doi: 10.1016/j.ecolmodel.2011.07.002.

[16]

T. GalochkinaM. Marion and V. Volpert, Initiation of reaction-diffusion waves of blood coagulation, Phys. D, 376-377 (2018), 160-170.  doi: 10.1016/j.physd.2017.11.006.

[17]

G. F. Gause, The Struggle for Existence, Williams and Wilkins, Baltimore, 1934. doi: 10.5962/bhl.title.4489.

[18]

S. GenieysN. Bessonov and V. Volpert, Mathematical model of evolutionary branching, Math. Comput. Modelling, 49 (2009), 2109-2115.  doi: 10.1016/j.mcm.2008.07.018.

[19]

S. GenieysV. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Model. Nat. Phenom., 1 (2006), 65-82.  doi: 10.1051/mmnp:2006004.

[20]

S. GenieysV. Volpert and P. Auger, Adaptive dynamics: Modelling Darwin's divergence principle, Comp. Ren. Biol., 329 (2006), 876-879.  doi: 10.1016/j.crvi.2006.08.006.

[21]

J. D. Murray, Mathematical Biology. Ⅱ: Spatial Models And Biomedical Applications, Interdisciplinary Applied Mathematics, 19, Springer-Verlag, New York, 2003. doi: 10.1007/b98869.

[22]

S. PalS. Ghorai and M. Banerjee, Analysis of a prey-predator model with non-local interaction in the prey population, Bull. Math. Biol., 80 (2018), 906-925.  doi: 10.1007/s11538-018-0410-x.

[23]

S. V. Petrovskii and H. Malchow, A minimal model of pattern formation in a prey-predator system, Math. Comput. Modelling, 29 (1999), 49-63.  doi: 10.1016/S0895-7177(99)00070-9.

[24]

J. A. SherrattB. T. Eagan and M. A. Lewis, Oscillations and chaos behind predator-prey invasion: Mathematical artifact or ecological reality?, Phil. Trans. R. Soc. Lond. B, 352 (1997), 21-38.  doi: 10.1098/rstb.1997.0003.

[25]

V. Volpert, Branching and aggregation in self-reproducing systems, in MMCS, Mathematical Modelling of Complex Systems, ESAIM Proc. Surveys, 47, EDP Sci., Les Ulis, 2014,116–129. doi: 10.1051/proc/201447007.

[26]

V. Volpert, Elliptic Partial Differential Equations, Monographs in Mathematics, 104, Birkhäuser/Springer Basel AG, Basel, 2014. doi: 10.1007/978-3-0348-0813-2.

[27]

V. Volpert, Pulses and waves for a bistable nonlocal reaction-diffusion equation, Appl. Math. Lett., 44 (2015), 21-25.  doi: 10.1016/j.aml.2014.12.011.

Figure 1.  Single pulse solution for prey and predator population for $ b = 10, d_1 = 0.1, d_2 = 0.1 $
Figure 2.  Moving pulse for $ b = 10, d_1 = 0.3, d_2 = 0.1 $ (a) after time $ t = 500 $; (b) after time $ t = 600 $; (c) $ x $-$ t $ profile for $ L = 20 $
Figure 3.  For two bifurcation diagrams, $ a = 1 $, $ \sigma_1 = 0.1 $, $ k_1 = k_2 = 0.35 $, $ \sigma_2 = 0.2 $ are fixed. (a) Bifurcation diagram in $ (L,d_1) $-plane for the values of parameters $ b = 5 $, $ \alpha = \beta = 0.35 $, $ d_2 = 0.1 $. (b) Bifurcation diagram in $ (d_1,b) $-plane for the values of other parameters $ \alpha = \beta = 0.363 $, $ d_2 = 1 $
Figure 4.  Periodic travelling wave in the case of nonlocal consumption followed by a spatio-temporal structure. Left: a snapshot of solution with prey (blue) and predator (red) distributions. Right: level lines of the prey distribution $ u(x,t) $. The values of other parameters are $ a = 1,b = 1, \alpha = \beta = 0.363, \sigma_1 = 0.1, k_1 = k_2 = 0.35, \sigma_2 = 0.2, d_1 = d_2 = 1 $
Figure 5.  Multiple moving pulses in the case of nonlocal consumption. Left: a snapshot of solution with prey (blue) and predator (red) distributions. Right: level lines of the prey distribution $ u(x,t) $. The values of other parameters are $ a = 1,b = 1, \alpha = \beta = 0.375, \sigma_1 = 0.1, k_1 = k_2 = 0.35, \sigma_2 = 0.2, d_1 = d_2 = 1 $
Figure 6.  Different regimes observed in the case of nonlocal consumption presented on the $ (d_1,N) $ parameter plane for $ \alpha = 0.35 $ (left) and $ (N,\alpha) $ parameter plane for $ d_1 = 1 $ (right). The values of other parameters are $ a = 1,b = 1,\sigma_1 = 0.1, k_1 = k_2 = 0.35, \sigma_2 = 0.2, d_2 = 1 $
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