A prey‐predator model with diffusion and a supplementary resource for the prey in a two‐patch environmentFootnote¹

This work partially was carried out at Department of Mathematics, Indian Institute of Technology, Kanpur‐208016, India. Author is thankful to Prof. J. B. Shukla for his valuable suggestion.

References

• Ballyk , M.M. and Wolkowicz , G.S.K. 1995 . An examination of the thresholds of enrichment: A resource‐based growth model . J. Math. Biol , 33 : 435 – 457 .
   Web of Science ®Google Scholar
• Butler , G.J. and Wolkowicz , G.S.K. 1985 . A mathematical model of the chemostat with a general class of functions describing nutrient uptake . SIAMJ.Appl. Math , 45 (1) : 125 – 138 .
   Google Scholar
• Dhar , J. 2003 . Modelling and analysis: the effect of industralization on diffusive forest resource biomass in closed habitat . J. Appl. Math., African Diaspora Journal of Math , 2 (1) : 142 – 159 .
   Google Scholar
• Dhar , J. and Shukla , J.B. 2000 . “ A single species model with diffusion and harvesting in a two‐patch habitat ” . In Mathematical Analysis and Application, chapter 9, , New Delhi : Narosa Publishing House .
   Google Scholar
• Dhar , J. and Singh , H. 2003 . Steady state distribution and stability behavior of a single species population with diffusion in n‐patch habitat . Far EastJ.Appl. Math , 11 (2) : 103 – 119 .
   Google Scholar
• Freedman , H.I. and Shukla , J.B. 1989 . The effect of a predator resource on a diffusive predator‐prey system . Natural Resources Modeling , 3 (3) : 359 – 383 .
   Google Scholar
• Freedman , H.I. and Wu , J. 1992 . Steady‐state analysis in a model for population diffusion in a multi‐patch environment . Nonl. Anal. Theory Methods & Appl , 18 (6) : 517 – 542 . [8]
   Web of Science ®Google Scholar
• Gopalsamy , K. 1977 . Competition, dispersion and co‐existence . Math. Bio. Science , 33 : 25 – 33 .
   Web of Science ®Google Scholar
• Gopalsamy , K. 1986 . Convergence in a resource‐based competition system . Bull. Math. Biol , 48 (5/6) : 681 – 699 .
   Web of Science ®Google Scholar
• Hallam , T.G. 1979 . A temporal study of diffusion effects on a population modelled by quadratic growth . Nonl. Anal. Theory Methods & Appl , 3 : 123 – 133 .
   Google Scholar
• Herbert, Elsworth , D. and Telling , R.C. 1956 . The continuous culture of bacteria: a theoretical and experimental study . J. Gen. Microbiology , 4 : 601 – 622 .
   Google Scholar
• Hsu , S.B. 1981 . On a resource based ecological competition model with interference . J. Math. Biol , 12 : 45 – 52 .
   Google Scholar
• Hsu , S.B. , Hubbell , S.P. and Waltman , P. 1977 . A mathematical theory of single‐nutrient competitions in continuous cultures of microorganism . SIAMJ.Appl. Math , 32 : 366 – 383 .
   Web of Science ®Google Scholar
• Kot , M. 2001 . Elements of Mathematical Ecology , Cambridge : Cambridge University Press .
   Google Scholar
• Levin , S.A. 1974 . Dispersion and population interactions . Am. Nat , 108 : 207 – 228 .
   Web of Science ®Google Scholar
• Levin , S.A. 1976 . Spatial patterning and the structure of ecological communities . Lectures on Mathematics in the Life Sciences , 8
   Google Scholar
• Li , B. and Smith , H. 2001 . How many species can two essential resources support? . SIAMJ.Appl. Math , 62 (1) : 336 – 366 .
   Web of Science ®Google Scholar
• McMurtrie , R. 1978 . Persistence and stability of single species and prey‐predator system in spatially heterogeneous environments . Math. Biosc , 39 : 11 – 51 .
   Web of Science ®Google Scholar
• Mitra , D. , Mukherjee , D. and Roy , A.B. 1992 . Permanent coexistence in a resource‐based competition system . Ecol. Model , 60 : 77 – 85 .
   Web of Science ®Google Scholar
• Munn , R.E. and Fedorov , V. 1984 . “ The environmental assessment ” . In IIASA Project Report, Vol.1, , Vol.1 , Laxenburg, Austria : International Institute for Applied Systems Analysis .
   Google Scholar
• Nallaswamy , R. and Shukla , J.B. 1982 . Effects of dispersal on the stability of a Prey‐Predator system with functional response . Math. Biosc , 60 : 123 – 132 .
   Web of Science ®Google Scholar
• Novick , A. and Szilard , L. 1950 . Description of the chemostat . Science , 112 : 715 – 716 .
   PubMed Web of Science ®Google Scholar
• Okubo , A. Springer‐Verlag 1980 . Diffusion and Ecological Problem: Mathematical models Springer‐Verlag , New York
   Google Scholar
• Pao , C.V. 1981 . Coexistence and stability of a competition diffusion system in population dynamics . J. Math. Anal. Appl , 83 : 54 – 76 .
   Google Scholar
• Rothe , F. Springer‐Verlag 1984 . Global Solution of Reaction‐Diffusion Systems , Springer‐Verlag , Berlin Heidelberg New York Tokyo : Lecture Notes in Math. 1072 .
   Google Scholar
• Shukla , J.B. , Freedman , H.I. , Pal , V.N. , Mishra , O.P. , Agarwal , M. and Shukla , A. 1989 . Degradation and subsequent regeneration of aresource: A mathematical model . Ecol. Model , 44 : 219 – 229 .
   Web of Science ®Google Scholar
• Shukla , V.P. and Shukla , J.B. 1982 . Multispecies food webs with diffusion . J. Math. Biol , 13 : 339 – 344 .
   Google Scholar
• Williams , F.M. 1971 . “ Dynamics of microbial populations ” . In Systems Analysis and Simulation in Ecology, Chapter‐3, , New York : Academic Press .
   Google Scholar
• This work partially was carried out at Department of Mathematics, Indian Institute of Technology, Kanpur‐208016, India. Author is thankful to Prof. J. B. Shukla for his valuable suggestion.
   Google Scholar