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June  2018, 15(3): 775-805. doi: 10.3934/mbe.2018035

Pattern formation of a predator-prey model with the cost of anti-predator behaviors

Department of Applied Mathematics, University of Western Ontario, London, ON, Canada N6A 5B7

* Corresponding author: Xingfu Zou.

Current address: Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, Canada

Received  September 21, 2016 Revised  June 07, 2017 Published  December 2017

Fund Project: Research partially supported by the Natural Sciences and Engineering Research Council of Canada.

We propose and analyse a reaction-diffusion-advection predator-prey model in which we assume that predators move randomly but prey avoid predation by perceiving a repulsion along predator density gradient. Based on recent experimental evidence that anti-predator behaviors alone lead to a 40% reduction on prey reproduction rate, we also incorporate the cost of anti-predator responses into the local reaction terms in the model. Sufficient and necessary conditions of spatial pattern formation are obtained for various functional responses between prey and predators. By mathematical and numerical analyses, we find that small prey sensitivity to predation risk may lead to pattern formation if the Holling type Ⅱ functional response or the Beddington-DeAngelis functional response is adopted while large cost of anti-predator behaviors homogenises the system by excluding pattern formation. However, the ratio-dependent functional response gives an opposite result where large predator-taxis may lead to pattern formation but small cost of anti-predator behaviors inhibits the emergence of spatial heterogeneous solutions.

Citation: Xiaoying Wang, Xingfu Zou. Pattern formation of a predator-prey model with the cost of anti-predator behaviors. Mathematical Biosciences & Engineering, 2018, 15 (3) : 775-805. doi: 10.3934/mbe.2018035
References:
[1]

B. E. AinsebaM. Bendahmane and A. Noussair, A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Analysis: Real World Applications, 9 (2008), 2086-2105.  doi: 10.1016/j.nonrwa.2007.06.017.

[2]

N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, Journal of Differential Equations, 33 (1979), 201-225.  doi: 10.1016/0022-0396(79)90088-3.

[3]

D. AlonsoF. Bartumeus and J. Catalan, Mutual interference between predators can give rise to Turing spatial patterns, Ecology, 83 (2002), 28-34. 

[4]

H. Amann, Dynamic theory of quasilinear parabolic equations Ⅱ: Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. 

[5]

H. Amann, Dynamic theory of quasilinear parabolic systems Ⅲ: Global existence, Mathematische Zeitschrift, 202 (1989), 219-250.  doi: 10.1007/BF01215256.

[6]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, 133 (1993), 9-126.  doi: 10.1007/978-3-663-11336-2_1.

[7]

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.

[8]

V. N. BiktashevJ. BrindleyA. V. Holden and M. A. Tsyganov, Pursuit-evasion predator-prey waves in two spatial dimensions, Chaos: An Interdisciplinary Journal of Nonlinear Science, 14 (2004), 988-994.  doi: 10.1063/1.1793751.

[9]

X. Cao, Y. Song and T. Zhang, Hopf bifurcation and delay-induced Turing instability in a diffusive lac operon model, International Journal of Bifurcation and Chaos, 26 (2016), 1650167, 22pp. doi: 10.1142/S0218127416501674.

[10]

S. Creel and D. Christianson, Relationships between direct predation and risk effects, Trends in Ecology & Evolution, 23 (2008), 194-201.  doi: 10.1016/j.tree.2007.12.004.

[11]

W. Cresswell, Predation in bird populations, Journal of Ornithology, 152 (2011), 251-263.  doi: 10.1007/s10336-010-0638-1.

[12]

D. L. DeAngelisR. A. Goldstein and R. V. O'neill, A model for tropic interaction, Ecology, 56 (1975), 881-892.  doi: 10.2307/1936298.

[13]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[14]

T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Advances in Applied Mathematics, 26 (2001), 280-301.  doi: 10.1006/aama.2001.0721.

[15]

C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, The Canadian Entomologist, 91 (1959), 293-320.  doi: 10.4039/Ent91293-5.

[16]

C. S. Holling, Some characteristics of simple types of predation and parasitism, The Canadian Entomologist, 91 (1959), 385-398.  doi: 10.4039/Ent91385-7.

[17]

H. Jin and Z. Wang, Global stability of prey-taxis systems, Journal of Differential Equations, 262 (2017), 1257-1290.  doi: 10.1016/j.jde.2016.10.010.

[18]

J. M. LeeT. Hillen and M. A. Lewis, Pattern formation in prey-taxis systems, Journal of Biological Dynamics, 3 (2009), 551-573.  doi: 10.1080/17513750802716112.

[19]

S. A. Levin, The problem of pattern and scale in ecology, Ecology, 73 (1992), 1943-1967. 

[20]

E. A. McGehee and E. Peacock-López, Turing patterns in a modified Lotka-Volterra model, Physics Letters A, 342 (2005), 90-98.  doi: 10.1016/j.physleta.2005.04.098.

[21]

A. B. MedvinskyS. V. PetrovskiiI. A. TikhonovaH. Malchow and B. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Review, 44 (2002), 311-370.  doi: 10.1137/S0036144502404442.

[22]

J. D. Meiss, Differential Dynamical Systems, SIAM, 2007. doi: 10.1137/1.9780898718232.

[23]

M. Mimura and J. D. Murray, On a diffusive prey-predator model which exhibits patchiness, Journal of Theoretical Biology, 75 (1978), 249-262.  doi: 10.1016/0022-5193(78)90332-6.

[24]

M. Mimura, Asymptotic behavior of a parabolic system related to a planktonic prey and predator system, SIAM Journal on Applied Mathematics, 37 (1979), 499-512.  doi: 10.1137/0137039.

[25]

A. MorozovS. Petrovskii and B. Li, Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect, Journal of Theoretical Biology, 238 (2006), 18-35.  doi: 10.1016/j.jtbi.2005.05.021.

[26]

J. D. Murray, Mathematical Biology Ⅱ: Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.

[27]

W. Ni and M. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, Journal of Differential Equations, 261 (2016), 4244-4274.  doi: 10.1016/j.jde.2016.06.022.

[28]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart, 10 (2002), 501-543. 

[29]

S. V. PetrovskiiA. Y. Morozov and E. Venturino, Allee effect makes possible patchy invasion in a predator-prey system, Ecology Letters, 5 (2002), 345-352.  doi: 10.1046/j.1461-0248.2002.00324.x.

[30]

D. Ryan and R. Cantrell, Avoidance behavior in intraguild predation communities: A cross-diffusion model, Discrete and Continuous Dynamical Systems, 35 (2015), 1641-1663.  doi: 10.3934/dcds.2015.35.1641.

[31]

H. Shi and S. Ruan, Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference, IMA Journal of Applied Mathematics, 80 (2015), 1534-1568.  doi: 10.1093/imamat/hxv006.

[32]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995.

[33]

Y. Song and X. Zou, Spatiotemporal dynamics in a diffusive ratio-dependent predator-prey model near a Hopf-Turing bifurcation point, Computers and Mathematics with Applications, 37 (2014), 1978-1997.  doi: 10.1016/j.camwa.2014.04.015.

[34]

Y. SongT. Zhang and Y. Peng, Turing-Hopf bifurcation in the reaction-diffusion equations and its applications, Communications in Nonlinear Science and Numerical Simulation, 33 (2016), 229-258.  doi: 10.1016/j.cnsns.2015.10.002.

[35]

Y. Song and X. Tang, Stability, steady-state bifurcations, and Turing patterns in a predator-prey model with herd behavior and prey-taxis, Studies in Applied Mathematics,(2017).

[36]

J. H. Steele, Spatial Pattern in Plankton Communities (Vol. 3), Springer Science & Business Media, 1978.

[37]

Y. Tao, Global existence of classical solutions to a predator-prey model with nolinear prey-taxis, Nonlinear Analysis: Real World Applications, 11 (2010), 2056-2064.  doi: 10.1016/j.nonrwa.2009.05.005.

[38]

J. WangJ. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, Journal of Differential Equations, 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004.

[39]

J. WangJ. Wei and J. Shi, Global bifurcation analysis and pattern formation in homogeneous diffusive predator-prey systems, Journal of Differential Equations, 260 (2016), 3495-3523.  doi: 10.1016/j.jde.2015.10.036.

[40]

X. WangL. Y. Zanette and X. Zou, Modelling the fear effect in predator-prey interactions, Journal of Mathematical Biology, 73 (2016), 1179-1204.  doi: 10.1007/s00285-016-0989-1.

[41]

X. WangW. Wang and G. Zhang, Global bifurcation of solutions for a predator-prey model with prey-taxis, Mathematical Methods in the Applied Sciences, 38 (2015), 431-443.  doi: 10.1002/mma.3079.

[42]

Z. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos: An Interdisciplinary Journal of Nonlinear Science, 17 (2007), 037108, 13 pp. doi: 10.1063/1.2766864.

[43]

S. WuJ. Shi and B. Wu, Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis, Journal of Differential Equations, 260 (2016), 5847-5874.  doi: 10.1016/j.jde.2015.12.024.

[44]

J. XuG. YangH. Xi and J. Su, Pattern dynamics of a predator-prey reaction-diffusion model with spatiotemporal delay, Nonlinear Dynamics, 81 (2015), 2155-2163.  doi: 10.1007/s11071-015-2132-z.

[45]

F. YiJ. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, Journal of Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.

[46]

L. Y. ZanetteA. F. WhiteM. C. Allen and M. Clinchy, Perceived predation risk reduces the number of offspring songbirds produce per year, Science, 334 (2011), 1398-1401.  doi: 10.1126/science.1210908.

[47]

T. Zhang and H. Zang, Delay-induced Turing instability in reaction-diffusion equations, Physical Review E, 90 (2014), 052908. doi: 10.1103/PhysRevE.90.052908.

show all references

References:
[1]

B. E. AinsebaM. Bendahmane and A. Noussair, A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Analysis: Real World Applications, 9 (2008), 2086-2105.  doi: 10.1016/j.nonrwa.2007.06.017.

[2]

N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, Journal of Differential Equations, 33 (1979), 201-225.  doi: 10.1016/0022-0396(79)90088-3.

[3]

D. AlonsoF. Bartumeus and J. Catalan, Mutual interference between predators can give rise to Turing spatial patterns, Ecology, 83 (2002), 28-34. 

[4]

H. Amann, Dynamic theory of quasilinear parabolic equations Ⅱ: Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. 

[5]

H. Amann, Dynamic theory of quasilinear parabolic systems Ⅲ: Global existence, Mathematische Zeitschrift, 202 (1989), 219-250.  doi: 10.1007/BF01215256.

[6]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, 133 (1993), 9-126.  doi: 10.1007/978-3-663-11336-2_1.

[7]

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.

[8]

V. N. BiktashevJ. BrindleyA. V. Holden and M. A. Tsyganov, Pursuit-evasion predator-prey waves in two spatial dimensions, Chaos: An Interdisciplinary Journal of Nonlinear Science, 14 (2004), 988-994.  doi: 10.1063/1.1793751.

[9]

X. Cao, Y. Song and T. Zhang, Hopf bifurcation and delay-induced Turing instability in a diffusive lac operon model, International Journal of Bifurcation and Chaos, 26 (2016), 1650167, 22pp. doi: 10.1142/S0218127416501674.

[10]

S. Creel and D. Christianson, Relationships between direct predation and risk effects, Trends in Ecology & Evolution, 23 (2008), 194-201.  doi: 10.1016/j.tree.2007.12.004.

[11]

W. Cresswell, Predation in bird populations, Journal of Ornithology, 152 (2011), 251-263.  doi: 10.1007/s10336-010-0638-1.

[12]

D. L. DeAngelisR. A. Goldstein and R. V. O'neill, A model for tropic interaction, Ecology, 56 (1975), 881-892.  doi: 10.2307/1936298.

[13]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[14]

T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Advances in Applied Mathematics, 26 (2001), 280-301.  doi: 10.1006/aama.2001.0721.

[15]

C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, The Canadian Entomologist, 91 (1959), 293-320.  doi: 10.4039/Ent91293-5.

[16]

C. S. Holling, Some characteristics of simple types of predation and parasitism, The Canadian Entomologist, 91 (1959), 385-398.  doi: 10.4039/Ent91385-7.

[17]

H. Jin and Z. Wang, Global stability of prey-taxis systems, Journal of Differential Equations, 262 (2017), 1257-1290.  doi: 10.1016/j.jde.2016.10.010.

[18]

J. M. LeeT. Hillen and M. A. Lewis, Pattern formation in prey-taxis systems, Journal of Biological Dynamics, 3 (2009), 551-573.  doi: 10.1080/17513750802716112.

[19]

S. A. Levin, The problem of pattern and scale in ecology, Ecology, 73 (1992), 1943-1967. 

[20]

E. A. McGehee and E. Peacock-López, Turing patterns in a modified Lotka-Volterra model, Physics Letters A, 342 (2005), 90-98.  doi: 10.1016/j.physleta.2005.04.098.

[21]

A. B. MedvinskyS. V. PetrovskiiI. A. TikhonovaH. Malchow and B. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Review, 44 (2002), 311-370.  doi: 10.1137/S0036144502404442.

[22]

J. D. Meiss, Differential Dynamical Systems, SIAM, 2007. doi: 10.1137/1.9780898718232.

[23]

M. Mimura and J. D. Murray, On a diffusive prey-predator model which exhibits patchiness, Journal of Theoretical Biology, 75 (1978), 249-262.  doi: 10.1016/0022-5193(78)90332-6.

[24]

M. Mimura, Asymptotic behavior of a parabolic system related to a planktonic prey and predator system, SIAM Journal on Applied Mathematics, 37 (1979), 499-512.  doi: 10.1137/0137039.

[25]

A. MorozovS. Petrovskii and B. Li, Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect, Journal of Theoretical Biology, 238 (2006), 18-35.  doi: 10.1016/j.jtbi.2005.05.021.

[26]

J. D. Murray, Mathematical Biology Ⅱ: Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.

[27]

W. Ni and M. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, Journal of Differential Equations, 261 (2016), 4244-4274.  doi: 10.1016/j.jde.2016.06.022.

[28]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart, 10 (2002), 501-543. 

[29]

S. V. PetrovskiiA. Y. Morozov and E. Venturino, Allee effect makes possible patchy invasion in a predator-prey system, Ecology Letters, 5 (2002), 345-352.  doi: 10.1046/j.1461-0248.2002.00324.x.

[30]

D. Ryan and R. Cantrell, Avoidance behavior in intraguild predation communities: A cross-diffusion model, Discrete and Continuous Dynamical Systems, 35 (2015), 1641-1663.  doi: 10.3934/dcds.2015.35.1641.

[31]

H. Shi and S. Ruan, Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference, IMA Journal of Applied Mathematics, 80 (2015), 1534-1568.  doi: 10.1093/imamat/hxv006.

[32]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995.

[33]

Y. Song and X. Zou, Spatiotemporal dynamics in a diffusive ratio-dependent predator-prey model near a Hopf-Turing bifurcation point, Computers and Mathematics with Applications, 37 (2014), 1978-1997.  doi: 10.1016/j.camwa.2014.04.015.

[34]

Y. SongT. Zhang and Y. Peng, Turing-Hopf bifurcation in the reaction-diffusion equations and its applications, Communications in Nonlinear Science and Numerical Simulation, 33 (2016), 229-258.  doi: 10.1016/j.cnsns.2015.10.002.

[35]

Y. Song and X. Tang, Stability, steady-state bifurcations, and Turing patterns in a predator-prey model with herd behavior and prey-taxis, Studies in Applied Mathematics,(2017).

[36]

J. H. Steele, Spatial Pattern in Plankton Communities (Vol. 3), Springer Science & Business Media, 1978.

[37]

Y. Tao, Global existence of classical solutions to a predator-prey model with nolinear prey-taxis, Nonlinear Analysis: Real World Applications, 11 (2010), 2056-2064.  doi: 10.1016/j.nonrwa.2009.05.005.

[38]

J. WangJ. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, Journal of Differential Equations, 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004.

[39]

J. WangJ. Wei and J. Shi, Global bifurcation analysis and pattern formation in homogeneous diffusive predator-prey systems, Journal of Differential Equations, 260 (2016), 3495-3523.  doi: 10.1016/j.jde.2015.10.036.

[40]

X. WangL. Y. Zanette and X. Zou, Modelling the fear effect in predator-prey interactions, Journal of Mathematical Biology, 73 (2016), 1179-1204.  doi: 10.1007/s00285-016-0989-1.

[41]

X. WangW. Wang and G. Zhang, Global bifurcation of solutions for a predator-prey model with prey-taxis, Mathematical Methods in the Applied Sciences, 38 (2015), 431-443.  doi: 10.1002/mma.3079.

[42]

Z. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos: An Interdisciplinary Journal of Nonlinear Science, 17 (2007), 037108, 13 pp. doi: 10.1063/1.2766864.

[43]

S. WuJ. Shi and B. Wu, Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis, Journal of Differential Equations, 260 (2016), 5847-5874.  doi: 10.1016/j.jde.2015.12.024.

[44]

J. XuG. YangH. Xi and J. Su, Pattern dynamics of a predator-prey reaction-diffusion model with spatiotemporal delay, Nonlinear Dynamics, 81 (2015), 2155-2163.  doi: 10.1007/s11071-015-2132-z.

[45]

F. YiJ. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, Journal of Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.

[46]

L. Y. ZanetteA. F. WhiteM. C. Allen and M. Clinchy, Perceived predation risk reduces the number of offspring songbirds produce per year, Science, 334 (2011), 1398-1401.  doi: 10.1126/science.1210908.

[47]

T. Zhang and H. Zang, Delay-induced Turing instability in reaction-diffusion equations, Physical Review E, 90 (2014), 052908. doi: 10.1103/PhysRevE.90.052908.

Figure 1.  Conditions of global stability of $E(\bar{u}, \bar{v})$ when $\alpha$ varies with $m(v) = m_1+m_2\, v$ and $p(u, v) = p.$ Parameters are: $r_0 = 5, \, a = 1, \, d = 0.2, \, p = 0.5, \, c = 0.5, \, m_1 = 0.3\, m_2 = 1, \, M = 10, d_u = 1, \, d_v = 2, \, k_0 = 1.$
Figure 2.  Conditions of global stability of $E(\bar{u}, \bar{v})$ when $k_0$ varies with $m(v) = m_1+m_2\, v$ and $p(u, v) = p.$ Parameters are: $r_0 = 5, \, a = 1, \, d = 0.2, \, p = 0.5, \, c = 0.5, \, m_1 = 0.3\, m_2 = 1, \, M = 10, d_u = 1, \, d_v = 2, \, \alpha = 0.5.$
Figure 3.  Spatial homogeneous steady states of $u, v$ with the Holling type Ⅱ functional response and density-dependent death function of predators when $\alpha$ is large. Parameters are: $r_0 =.8696, \, d =.1827, \, a =.6338, \, p =6.395, \, q =4.333, \, m_1 =0.72e-2, \, m_2 =.9816, \, c =.2645, \, d_u =0.2119e-1, \, d_v =1.531, \, \alpha =12, \, k_0 =0.1e-1, \, M =10, \, L = 4.$
Figure 4.  Spatial heterogeneous steady states of $u, v$ with the Holling type Ⅱ functional response and density-dependent death function of predators when $\alpha$ is small. Parameters are: $r_0 =.8696, \, d =.1827, \, a =.6338, \, p =6.395, \, q =4.333, \, m_1 =0.72e-2, \, m_2 =.9816, \, c =.2645, \, d_u =0.2119e-1, \, d_v =1.531, \, \alpha =8, \, k_0 =0.1e-1, \, M =10, \, L = 4.$
Figure 5.  Spatial homogeneous steady states of $u, v$ when $k_0 = 0, $ $\alpha$ is large, $m(v) = m_1, $ and $p(u, v) = b_1/(b_2\, v+u).$ Parameters are: $r_0 = 6.1885, \, d = 4.0730, \, a = 0.8481, \, b_1 = 4.5677, \, b_2 = 1.4380, \, m_1 = 1.6615, \, c = 0.9130, \, \alpha = 12, \, d_u = 0.0113, \, d_v = 4.7804, \, M = 10, \, k_0 = 0, \, L = 5.0212.$
Figure 6.  Spatial heterogenous steady states of $u, v$ when $k_0 = 0, $ $\alpha$ is small, $m(v) = m_1, $ and $p(u, v) = b_1/(b_2\, v+u).$ Parameters are: $r_0 = 6.1885, \, d = 4.0730, \, a = 0.8481, \, b_1 = 4.5677, \, b_2 = 1.4380, \, m_1 = 1.6615, \, c = 0.9130, \, \alpha = 5.1571, \, d_u = 0.0113, \, d_v = 4.7804, \, M = 10, \, k_0 = 0, \, L = 5.0212.$
Figure 7.  Conditions of diffusion-taxis-driven instability of $E(\bar{u}, \bar{v})$ with changing $\alpha$ when $k_0 \neq 0, $ $m(v) = m_1, $ and $p(u, v) = b_1/(b_2\, v+u).$ Parameters are: $r_0 =1.7939, \, d =0.2842, \, a =0.4373, \, b_1 =2.9354, \, b_2 =3.2998, \, m_1 =0.5614, \, c =0.6010, \, d_u =0.0344, \, d_v =7.2808, \, k_0 =8.0318, \, M = 10.$
Figure 8.  Spatial homogeneous steady states of $u,v$ when $k_0 \neq 0,$ $\alpha$ is small, $m(v)=m_1,$ and $p(u,v)=b_1/(b_2\,v+u).$ Parameters are: $r_0 = 1.7939,\, d = 0.2842,\, a = 0.4373,\, b_1 = 2.9354,\, b_2 = 3.2998,\, m_1 = 0.5614,\, c = 0.6010,\, d_u = 0.0344,\, d_v = 7.2808,\, k_0 = 8.0318,\, M=10,\,\alpha=0.3,\,L=2.6602.$
Figure 9.  Spatial heterogenous steady states of $u, v$ when $k_0 \neq 0, $ $\alpha$ is large, $m(v) = m_1, $ and $p(u, v) = b_1/(b_2\, v+u).$ Parameters are: $r_0 =1.7939, \, d =0.2842, \, a =0.4373, \, b_1 =2.9354, \, b_2 =3.2998, \, m_1 =0.5614, \, c =0.6010, \, d_u =0.0344, \, d_v =7.2808, \, k_0 =8.0318, \, M = 10, \, \alpha = 0.7957, \, L = 2.6602.$
Figure 10.  Spatial homogeneous steady states of $u, v$ when $k_0$ is small, $\alpha \neq 0, $ $m(v) = m_1, $ and $p(u, v) = b_1/(b_2\, v+u).$ Parameters are: $r_0 =1.7939, \, d =0.2842, \, a =0.4373, \, b_1 =2.9354, \, b_2 =3.2998, \, m_1 =0.5614, \, c =0.6010, \, d_u =0.0344, \, d_v =7.2808, \, k_0 =2, \, M = 10, \, \alpha = 0.7957, \, L = 2.6602.$
Figure 11.  Spatial homogeneous but temporal periodic solution $u, v$ over time when $m(v) = m_1+m_2\, v, $ $k_0$ is large, and $p(u, v) = b_1/(b_2\, v+u).$ Parameters are: $r_0 =4.8712, \, d =.9235, \, a =.9508, \, b_1 =.3433, \, b_2 =.6731, \, m_1 =0.228e-1, \, m_2 =.7908, \, c =.2959, \, d_u =.1516, \, d_v =8.5545, \, k_0 = 10, \, \alpha =7.4798, \, M = 10, \, L = 10.$
Figure 12.  Spatial homogeneous steady states of $u, v$ when $m(v) = m_1, k_0 \neq 0, $ $\alpha$ is large, and $p(u, v) = p/(1+q_1\, u+q_2\, v).$ Parameters are: $r_0 =.3558, \, d =0.832e-1, \, a =0.106e-1, \, p =.6313, \, q_1 =.4418,$ $q_2 =.3188, \, m_1 =.4901, \, c =.4780, \, d_u =0.324e-1, \, d_v =3.7446, \, M =100, \, \alpha = 0.1, \, k_0 = 1, L = 7$ .
Figure 13.  Spatial heterogeneous steady states of $u, v$ when $m(v) = m_1, k_0 \neq 0, $ $\alpha$ is small, and $p(u, v) = p/(1+q_1\, u+q_2\, v).$ Parameters are: $r_0 =.3558, \, d =0.832e-1, \, a =0.106e-1, \, p =.6313,$ $q_1 =.4418, \, q_2 =.3188, \, m_1 =.4901, \, c =.4780, \, d_u =0.324e-1, \, d_v =3.7446, \, M =100, \, \alpha = 0.01, \, k_0 = 1, L = 7$ .
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