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http://dx.doi.org/10.14477/jhm.2016.29.6.353
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A Historical Study on the Representations of Diffusion Phenomena in Mathematical Models for Population Changes of Biological Species |

Shim, Seong-A (Dept. of Math. Sungshin Women's University) |

Publication Information

Abstract

In mathematical population ecology which is an academic field that studies how populations of biological species change as times flows at specific locations in their habitats, PDE models have been studied in many aspects and found to have different properties from the classical ODE models. And different approaches to PDE type models in mathematical biology are still being tried currently. This article investigate various forms to express diffusion effects and review the history of PDE models involving diffusion terms in mathematical ecology. Semi-linear systems representing the spatial movements of each individual as random simple diffusion and quasi-linear systems describing more complex diffusions reflecting interspecific interactions are studied. Also it introduce a few of important problems to be solved in this field.

Keywords

population ecology; diffusion; semi-linear system; quasi-linear system;

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Times Cited By KSCI :
3 (Citation Analysis)

- Reference
- Cited By KSCI

1 | W. KO, I. AHN, Positive coexistence for a simple food chain model with ratiodependent functional response and cross-diffusion, Commun. Korean Math. Soc. 21(4) (2006), 701-717. DOI |

2 | A. KOLMOGOROV, I. PETROVSKII, N. PISCOUNOV, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, In V. M. TIKHOMIROV, editor, Selected Works of A. N. Kolmogorov I, Kluwer 1991, 248-270. Translated by V. M. Volosov from Bull. Moscow Univ., Math. Mech. 1 (1937), 1-25. |

3 | Y. KUANG, H. FREEDMAN, Uniqueness of limit cycles in Gause-type models of predator-prey system, Math. Biosci. 88 (1988), 67-84. DOI |

4 | K. KUTO, Stability of steady-state solutions to a prey-predator system with crossdiffusion, J. Differential Equations 197 (2004), 293-314. DOI |

5 | K. KUTO, Y. YAMADA, Multiple coexistence states for a prey-predator system with cross-diffusion, J. Differential Equations 197(2) (2004), 315-348. DOI |

6 | S. A. LEVIN, Dispersion and population interactions, Amer. Naturalist 108 (1974), 207-228. DOI |

7 | R. MAY, Stability and Complexity in Model Ecosystems, 2nd ed., Princeton Univ. press, Princeton, 1974. |

8 | M. MIMURA, T. NISHIDA, On a certain semilinear parabolic system related to Lotka-Volterra's ecological model, Publ. Research Inst. Math. Sci. Kyoto Univ. 14 (1978), 269-282. DOI |

9 | J. D. MURRAY, Non-existence of wave solutions for the class of reaction-diffusion eqautions given by the Volterra interacting-population equations with diffusion, J. Theor. Biol. 52 (1975), 459-469. DOI |

10 | A. OKUBO, L. A. LEVIN, Diffusion and Ecological Problems: modern perspective, Interdisciplinary Applied Mathematics, 2nd ed., Vol. 14, Springer, New York, 2001. |

11 | H. G. OTHMER, Current problems in pattern formation, Lectures on Mathematics in the Life Sciences 9 (1977), 57-85, S. A. Levin (ed), Amer. Math. Soc. |

12 | C. PAO, Strongly coupled elliptic systems and applications to Lotka-Volterra models with cross-diffusion, Nonlinear Analysis 60 (2005), 1197-1217. DOI |

13 | K. PEARSON, J. BLAKEMAN, Mathematical contributions to the theory of evolution - XV. A mathematical theory of random migration, Drapers' Company Research Mem. Biometric Series III, Dept. Appl. Math., Univ. College, Univ. London, 1906. |

14 | L. REAL, Ecological determinants of functioal response, Ecology 60 (1979), 481-485. DOI |

15 | K. RYU, I. AHN, Coexistence theorem of steady states for nonlinear self-cross diffusion system with competitive dynamics, J. Math. Anal. Appl. 283 (2003), 46-65. DOI |

16 | L. A. SEGEL, J. L. JACKSON, Dissipative structure: An explanation and an ecological example, J. Theor. Biol. 37 (1952), 545-559. |

17 | N. SHIESADA, K. KAWASAKI, E. TERAMOTO, Spatial segregation of interacting species, J. Theor. Biol. 79 (1978), 83-99. |

18 | S.-A. SHIM, Uniform Boundedness and Convergence of Solutions to Cross-Diffusion Systems, J. Differential Equations 185 (2002), 281-305. DOI |

19 | S.-A. SHIM, Long-time Properties of Prey-Predator System with Cross-Diffusion, Comm. KMS 21(2) (2006), 293-320. |

20 | S.-A. SHIM, Global Existence of Solutions to the Pre-Predator system with a Single Cross-Diffusion, Bull. KMS 43(2) (2006), 443-459. |

21 | S.-A. SHIM, -estimates on the prey-predator systems with cross-diffusion and functional responses, Comm. KMS 23(2) (2008), 211-227. |

22 | S.-A. SHIM, Mathematical models for population changes of two interacting species, The Korean Journal for History of Mathematics 25(1) (2012), 45-56. |

23 | J. G. SKELLAM, Random dispersal in theoretical populations, Biometrika 38 (1951), 196-218. DOI |

24 | J. G. SKELLA, Some philosophical aspects of mathematical modelling in empirical science with special reference to ecolgy, Mathematical Models in Ecology 13-28, J.N.R. Jeffers (ed.), London, Blackwell Sci. Publ., 1972. |

25 | J. G. SKELLAM, The formulation and interpretation of mathematical models of diffusion processes in population biology, Mathematical Theory of the Dynamics of Biological Populations, 63-85, M.S. Bartlett, R.W. Hiorns (eds.), New York, Academic press, 1973. |

26 | M. A. TSYGANOV, J. BRINDLEY, A. V. HOLDEN, V. N. BIKTASHEV, Soliton-like phenomena in one-dimensional cross-diffusion systems: a predator-prey pursuit and evasion example, Phys. D 197(1-2)(2004), 18-33. DOI |

27 | A. M. TURING, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. London B 237 (1952), 37-72. DOI |

28 | X. ZENG, Non-constant positive steady states of a prey-predator system with crossdiffusions, J. Math. Anal. Appl. 332(2) (2007), 989-1009. DOI |

29 | H. AMANN, Non-homogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992), 9-126, Teubner-Texte Math. 133, Teubner, Stuttgart, 1993. |

30 | E. AHMED, A. S. HEGAZI, A. S. ELGAZZAR, On persistence and stability of some biological systems with cross-diffusion, Advances in Complex Systems 7(1) (2004), 65-76. DOI |

31 | J. BROWNLEE, The mathematical theory of random migration and epidemic distribution, Proc. Roy. Soc. Edinburgh 31 (1911), 262-289. |

32 | H. S. CARSLAW, J. C. JAEGER, Conduction of Heat in Solids, 2nd ed., Oxford Univ. Press, 1959. |

33 | M. E. GURTIN, Some mahematical models for population dynamics that lead to segregation, Quart. J. Appli. Math. 32 (1974), 1-9. DOI |

34 | E. CONWAY, D. HOFF, J. SMOLLER, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math. 35 (1978), 1-16. DOI |

35 | R. A. FISHER, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937), 353-369. |

36 | H. I. FREEDMAN, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, 1980. |

37 | T. HILLEN, K. J. PAINTER, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology 58 (2009), 183-217. DOI |

38 | C. HOLLING, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can. 45 (1965), 3-60. |

39 | E. F. KELLER, L. A. SEGEL, Model for chemotaxis, J. Theor. Biol. 30 (1971), 225-234. DOI |

40 | M. G. KENDALL, A form of wave propagation associated with the equation of heat conduction, Proc. Cambridge Phil. Soc. 44 (1948), 591-593. DOI |

41 | K. KISHMOTO, The diffusive Lotka-Volterra system with three species can have a stable, non-constant equilibrium solution, J. Math. Biol. 16 (1982), 103-112. DOI |

42 | K. KISHIMOTO, M. MIMURA, K. YOSHIDA, Stable spatio-temporal oscillations of diffusive Lotka-Volterra system with 3 or more species, J. Math. Biol. 18 (1983), 213-221. DOI |