An exact solution of a diffusive predator-prey system. (English) Zbl 1145.92341

Summary: We consider a system of two nonlinear partial differential equations describing the spatio-temporal dynamics of a predator-rey community where the prey per capita growth rate is damped by the Allee effect. Using an appropriate change of variables, we obtain an exact solution of the system, which appears to be related to the issue of biological invasion. In the large-time limit, or for appropriate parameter values, this solution describes the propagation of a travelling population front. We show that the properties of the solution exhibit biologically reasonable dependence on the parameter values; in particular, it predicts that the travelling front of invasive species can be stopped or reversed owing to the impact of predation.


92D40 Ecology
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35K57 Reaction-diffusion equations
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[1] Ablowitz, M.J. & Zeppetella, A. 1979 Explicit solutions of Fisher’s equations for a special wave speed. Bull. Math. Biol. <b>41</b>, 835–840. · Zbl 0423.35079
[2] Allee, W.C. 1938 The social life of animals. New York: Norton and Co.
[3] Aronson, D.G. & Weinberger, H.F. 1978 Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. <b>30</b>, 33–76. · Zbl 0407.92014
[4] Berezovskaya, F.S. & Karev, G.P. 1999 Bifurcations of travelling waves in population taxis models. Physics–Uspekhi <b>169</b>, 1011–1024.
[5] Calogero, F. & Xiaoda, J. 1991 C-integrable nonlinear partial differential equations. J. Math. Phys. <b>32</b>, 875–887. · Zbl 0768.35071
[6] Campos, D., Fort, J. & Llebot, J.E. 2002 Reaction-diffusion wave fronts: multigeneration biological species under climate change. Phys. Rev. E <b>66</b>, 062901(4).
[7] Cannas, S.A., Marco, D.E. & Paez, S.A. 2003 Modelling biological invasions: species traits, species interactions, and habitat heterogeneity. Math. Biosci. <b>183</b>, 93–110. · Zbl 1011.92043
[8] Dennis, B. 1989 Allee effects: population growth, critical density, and the chance of extinction. Nat. Res. Model. <b>3</b>, 481–538. · Zbl 0850.92062
[9] Dubois, D. 1975 A model of patchiness for prey–predator plankton populations. Ecol. Model. <b>1</b>, 67–80.
[10] Dunbar, S.R. 1983 Travelling wave solutions of diffusive Lotka–Volterra equations. J. Math. Biol. <b>17</b>, 11–32. · Zbl 0509.92024
[11] Dunbar, S.R. 1986 Travelling waves in diffusive predator–prey equations: periodic orbits and point-to-periodic heteroclinic orbits. SIAM J. Appl. Math. <b>46</b>, 1057–1078. · Zbl 0617.92020
[12] Edwards, A.M. & Yool, A. 2000 The role of higher predation in plankton population models. J. Plankt. Res. <b>22</b>, 1085–1112.
[13] Fagan, W.F. & Bishop, J.G. 2000 Trophic interactions during primary succession: herbivores slow a plant reinvasion at Mount St. Helens. Am. Nat. <b>155</b>, 238–251.
[14] Fagan, W.F., Lewis, M.A., Neubert, M.G. & van den Driessche, P. 2002 Invasion theory and biological control. Ecol. Lett. <b>5</b>, 148–157.
[15] Frantzen, J. & van den Bosch, F. 2000 Spread of organisms: can travelling and dispersive waves be distinguished?. Basic Appl. Ecol. <b>1</b>, 83–91.
[16] Fisher, R. 1937 The wave of advance of advantageous genes. Ann. Eugen. <b>7</b>, 355–369. · JFM 63.1111.04
[17] Hadeler, K.P. & Rothe, F. 1975 Travelling fronts in nonlinear diffusion equations. J. Math. Biol. <b>2</b>, 251–263. · Zbl 0343.92009
[18] Holmes, E.E., Lewis, M.A., Banks, J.E. & Veit, R.R. 1994 Partial differential equations in ecology: spatial interactions and population dynamics. Ecology <b>75</b>, 17–29.
[19] Hopf, E. 1950 The partial differential equation u<sub>t</sub>+uu<sub>x</sub>={\(\mu\)}u<sub>xx</sub>. Comm. Pure Appl. Math. <b>3</b>, 201–216.
[20] Kawahara, T. & Tanaka, M. 1983 Interactions of travelling fronts: an exact solution of a nonlinear diffusion equation. Phys. Lett. A <b>97</b>, 311–314.
[21] Kolmogorov, A.N., Petrovsky, I.G. & Piskunov, N.S. 1937 Investigation of the equation of diffusion combined with increasing of the substance and its application to a biology problem. Bull. Moscow State Univ. Ser. A: Math. Mech. <b>1</b>, (6), 1–25.
[22] Kot, M., Lewis, M.A. & van der Driessche, P. 1996 Dispersal data and the spread of invading organisms. Ecology <b>77</b>, 2027–2042.
[23] Kudryashov, N.A. 1993 Singular manifold equations and exact solutions for some nonlinear partial differential equations. Phys. Lett. A <b>182</b>, 356–362.
[24] Larson, D.A. 1978 Transient bounds and time-asymptotic behavior of solutions to nonlinear equations of Fisher type. SIAM J. Appl. Math. <b>34</b>, 93–103. · Zbl 0373.35036
[25] Leach, J.A., Needham, D.J. & Kay, A.L. 2002 The evolution of reaction-diffusion waves in a class of scalar reaction-diffusion equations: algebraic decay rates. Physica D <b>167</b>, 153–182. · Zbl 1002.35068
[26] Lewis, M.A. & Kareiva, P. 1993 Allee dynamics and the spread of invading organisms. Theor. Popul. Biol. <b>43</b>, 141–158. · Zbl 0769.92025
[27] Lotka, A.J. 1925 Elements of physical biology. Baltimore: Williams and Wilkins. · JFM 51.0416.06
[28] Malchow, H. & Petrovskii, S.V. 2002 Dynamical stabilization of an unstable equilibrium in chemical and biological systems. Math. Comp. Model. <b>36</b>, 307–319. · Zbl 1021.92026
[29] Matsuda, H., Ogita, N., Sasaki, A. & Sato, K. 1992 Statistical mechanics of population. Progr. Theor. Phys. <b>88</b>, 1035–1049.
[30] Morozov, A.Y., Petrovskii, S.V. & Li, B.-L. 2004 Bifurcations and chaos in a predator–prey system with the Allee effect. Proc. R. Soc. B <b>271</b>, 1407–1414.
[31] Murray, J.D. 1989 Mathematical biology. Berlin: Springer. · Zbl 0682.92001
[32] Murray, J.D. 1993 Mathematical biology. 2nd edn. Berlin: Springer. · Zbl 0779.92001
[33] Newell, A.C. 1985 Solitons in mathematics and physics. Philadelphia: SIAM. · Zbl 0565.35003
[34] Newell, A.C. & Whitehead, J.A. 1969 Finite bandwidth, finite amplitude convection. J. Fluid Mech. <b>38</b>, 279–303. · Zbl 0187.25102
[35] Newman, W.I. 1980 Some exact solutions to a nonlinear diffusion problem in population genetics and combustion. J. Theor. Biol. <b>85</b>, 325–334.
[36] Ognev, M.V., Petrovskii, S.V. & Prostokishin, V.M. 1995 Dynamics of formation of a switching wave in a dissipative bistable medium. Tech. Phys. <b>40</b>, 521–524.
[37] Okubo, A. 1980 Diffusion and ecological problems: mathematical models. Berlin: Springer. · Zbl 0422.92025
[38] Owen, M.R. & Lewis, M.A. 2001 How predation can slow, stop or reverse a prey invasion. Bull. Math. Biol. <b>63</b>, 655–684. · Zbl 1323.92181
[39] Petrovskii, S.V. 1999 Exact solutions of the forced Burgers equation. Tech. Phys. <b>44</b>, 181–187.
[40] Petrovskii, S.V. & Li, B.-L. 2003 An exactly solvable model of population dynamics with density-dependent migrations and the Allee effect. Math. Biosci. <b>186</b>, 79–91. · Zbl 1027.92026
[41] Petrovskii, S.V. & Malchow, H. 2000 Critical phenomena in plankton communities: KISS model revisited. Nonlinear Anal.: Real World Appl. <b>1</b>, 37–51. · Zbl 0996.92037
[42] Petrovskii, S.V. & Shigesada, N. 2001 Some exact solutions of a generalized Fisher equation related to the problem of biological invasion. Math. Biosci. <b>172</b>, 73–94. · Zbl 0983.92031
[43] Petrovskii, S.V., Kawasaki, K., Takasu, F. & Shigesada, N. 2001 Diffusive waves, dynamical stabilization and spatio-temporal chaos in a community of three competitive species. Jpn. J. Ind. Appl. Math. <b>18</b>, 459–481. · Zbl 0983.35065
[44] Petrovskii, S. V. Morozov, A. Y. & Li, B.-L. In press. Regimes of biological invasion in a predator–prey system with the Allee effect. Bull. Math. Biol. · Zbl 1334.92363
[45] Sato, K., Matsuda, H. & Sasaki, A. 1994 Pathogen invasion and host extinction in lattice structured populations. J. Math. Biol. <b>32</b>, 251–268. · Zbl 0790.92022
[46] Savill, N.J. & Hogeweg, P. 1999 Competition and dispersal in predator–prey waves. Theor. Popul. Biol. <b>56</b>, 243–263. · Zbl 0965.92029
[47] Sherratt, J.A. 2001 Periodic travelling waves in cyclic predator–prey systems. Ecol. Lett. <b>4</b>, 30–37.
[48] Sherratt, J. & Marchant, B.P. 1996 Algebraic decay and variable speeds in wavefront solutions of a scalar reaction-diffusion equation. IMA J. Appl. Math. <b>56</b>, 289–302. · Zbl 0858.35062
[49] Sherratt, J.A., Lewis, M.A. & Fowler, A.C. 1995 Ecological chaos in the wake of invasion. Proc. Natl Acad. Sci. USA <b>92</b>, 2524–2528. · Zbl 0819.92024
[50] Shigesada, N. & Kawasaki, K. 1997 Biological invasions: theory and practice. Oxford: Oxford University Press.
[51] Skellam, J.G. 1951 Random dispersal in theoretical populations. Biometrika <b>38</b>, 196–218. · Zbl 0043.14401
[52] Steele, J.H. & Henderson, E.W. 1992 The role of predation in plankton model. J. Plankt. Res. <b>14</b>, 157–172.
[53] Volpert, A.I., Volpert, V.A. & Volpert, V.A. 1994 Travelling wave solutions of parabolic systems. Providence, PA: American Mathematical Society. · Zbl 0805.35143
[54] Volterra, V. 1926 Fluctuations in the abundance of a species considered mathematically. Nature <b>118</b>, 558–560. · JFM 52.0453.03
[55] Wang, M.-E. & Kot, M. 2001 Speeds of invasion in a model with strong or weak Allee effects. Math. Biosci. <b>171</b>, 83–97. · Zbl 0978.92033
[56] Yachi, S., Kawasaki, K., Shigesada, N. & Teramoto, E. 1989 Spatial patterns of propagating waves of fox rabies. Forma <b>4</b>, 3–12.
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