A. Alonso-Izquierdo 1, M.A. González León 1, M. de la Torre Mayado 2

Transcription

1 A Holling type II -Izquierdo 1, M.A. González León 1, Mayado 2 1 Departamento de Matemática Aplicada Universidad de Salamanca) and UIC MathPhys-CyL 2 Departamento de Física Fundamental Universidad de Salamanca) and UIC MathPhys-CyL Workshop: Nonlinear Integrable Systems, Burgos 2016

2 POPULATION DYNAMICS: OUTLINE This talk was presented in the workshop Nonlinear Integrable systems to honour Professor Orlando Ragnisco in his 70th anniversary celebrated in Burgos). In this work we construct a mathematical to describe the interaction between Pareas iwasakii snake predator and dextral and sinistral Satsuma prey populations. The ecological features of these species are beautifully introduced in the references [1-4], whose graphical support graphics and videos) are used in this presentation.

3 POPULATION DYNAMICS: OUTLINE

4 POPULATION 1: DEXTRAL SATSUMA SNAIL Pareas iwasakii is a snail-eating specialist, even newly hatched individuals feed on snails. It has asymmetric jaws, which facilitates feeding on snails with dextral shells. Lengh: cm). Satsuma is a genus of air-breathing land snails, terrestrial pulmonate gastropod mollusks in the family Camaenidae. They are with dextral clockwise coiled) shells. Shell diameter 2.8 cm.) Kingdom: Phylum: Subphylum: Class: Order: Suborder: Family: Subfamily: Genus: Species: Animalia Chordata Vertebrata Reptilia Squamata Serpentes Colubridae Pareatinae Pareas P. iwasakii Kingdom: Phylum: Class: Superfamily: Family: Genus: Animalia Mollusca Gastropoda Helicoidea Camaenidae Satsuma

5 POPULATION 2: PAREAS IWASAKII SNAKE Pareas iwasakii is a snail-eating specialist, even newly hatched individuals feed on snails. It has asymmetric jaws, which facilitates feeding on snails with dextral shells. Lengh: cm). Satsuma is a genus of air-breathing land snails, terrestrial pulmonate gastropod mollusks in the family Camaenidae. They are with dextral clockwise coiled) shells. Shell diameter 2.8 cm.) Kingdom: Phylum: Subphylum: Class: Order: Suborder: Family: Subfamily: Genus: Species: Animalia Chordata Vertebrata Reptilia Squamata Serpentes Colubridae Pareatinae Pareas P. iwasakii Kingdom: Phylum: Class: Superfamily: Family: Genus: Animalia Mollusca Gastropoda Helicoidea Camaenidae Satsuma

6 A PREDATOR-PREY MODEL X 1 t): Satsuma population Yt): Pareas iwasakii population dx 1 Prey logistic growth term = rx 1 1 X ) 1 K Number of hunted preys per time unit δx 1 Y dy = sy + β δx 1 Y Predator growth term Predator growth term in absence of prey due to predation

7 A PREDATOR-PREY MODEL X 1 t): Satsuma population Yt): Pareas iwasakii population dx 1 Prey logistic growth term = rx 1 1 X ) 1 K Number of hunted preys per time unit δx 1 Y dy = sy + β δx 1 Y Predator growth term Predator growth term in absence of prey due to predation

8 A PREDATOR-PREY MODEL X 1 t): Satsuma population Yt): Pareas iwasakii population dx 1 Prey logistic growth term = rx 1 1 X ) 1 K Number of hunted preys per time unit δx 1 Y dy = sy + β δx 1 Y Predator growth term Predator growth term in absence of prey due to predation

9 A LOTKA-VOLTERRA PREDATOR-PREY MODEL X 1 t): Satsuma population Yt): Pareas iwasakii population dx 1 Prey logistic growth term = rx 1 1 X ) 1 K Number of hunted preys per time unit α X 1 Y dy = sy + β X 1 Y Predator growth term Predator growth term in absence of prey due to predation

10 A HOLLING TYPE II PREDATOR-PREY MODEL X 1 t): Satsuma population Yt): Pareas iwasakii population dx 1 Prey logistic growth term = rx 1 1 X ) 1 K Number of hunted preys per time unit ax 1 Y 1 + ex 1 dy = sy + b X 1Y 1 + ex 1 Predator growth term in absence of prey Predator growth term due to predation

11 A HOLLING TYPE II PREDATOR-PREY MODEL X 1 t): Satsuma population dx 1 = rx 1 1 X ) 1 K dy Yt): Pareas iwasakii population ax 1 Y 1 + ex 1 = sy + b X 1Y 1 + ex 1

12 A HOLLING TYPE II PREDATOR-PREY MODEL X 1 t): Satsuma population dx 1 = rx 1 1 X ) 1 K dy Yt): Pareas iwasakii population ax 1 Y 1 + ex 1 = sy + b X 1Y 1 + ex 1 non-dimensional variables τ = rt x 1 = X 1 /K y = ay/r ɛ = Ke i σ = s/r

13 A HOLLING TYPE II PREDATOR-PREY MODEL X 1 t): Satsuma population dx ) 1 = x 1 1 x 1 dy = β σ)y Yt): Pareas iwasakii population x 1 y 1 + ɛx 1 β y 1 + ɛx 1 non-dimensional variables τ = rt x 1 = X 1 /K y = ay/r ɛ = Ke i σ = s/r

14 A HOLLING TYPE II PREDATOR-PREY MODEL X 1 t): Satsuma population dx ) 1 = x 1 1 x 1 dy = β σ)y Yt): Pareas iwasakii population x 1 y 1 + ɛx 1 β y 1 + ɛx 1 non-dimensional variables τ = rt x 1 = X 1 /K y = ay/r ɛ = Ke i σ = s/r Case A β + σ β σ > ɛ Case B β + σ β σ < ɛ

15 A HOLLING TYPE II PREDATOR-PREY MODEL X 1 t): Satsuma population dx ) 1 = x 1 1 x 1 dy = β σ)y Yt): Pareas iwasakii population x 1 y 1 + ɛx 1 β y 1 + ɛx 1 non-dimensional variables τ = rt x 1 = X 1 /K y = ay/r ɛ = Ke i σ = s/r Case A β + σ β σ > ɛ Case B β + σ β σ < ɛ

16 A HOLLING TYPE II PREDATOR-PREY MODEL X 1 t): Satsuma population dx ) 1 = x 1 1 x 1 dy = β σ)y Yt): Pareas iwasakii population x 1 y 1 + ɛx 1 β y 1 + ɛx 1 non-dimensional variables τ = rt x 1 = X 1 /K y = ay/r ɛ = Ke i σ = s/r Case A β + σ β σ > ɛ Case B β + σ β σ < ɛ

17 A HOLLING TYPE II PREDATOR-PREY MODEL X 1 t): Satsuma population dx ) 1 = x 1 1 x 1 dy = β σ)y Yt): Pareas iwasakii population x 1 y 1 + ɛx 1 β y 1 + ɛx 1 non-dimensional variables τ = rt x 1 = X 1 /K y = ay/r ɛ = Ke i σ = s/r Case A β + σ β σ > ɛ Case B β + σ β σ < ɛ

18 A HOLLING TYPE II PREDATOR-PREY MODEL X 1 t): Satsuma population dx ) 1 = x 1 1 x 1 dy = β σ)y Yt): Pareas iwasakii population x 1 y 1 + ɛx 1 β y 1 + ɛx 1 non-dimensional variables τ = rt x 1 = X 1 /K y = ay/r ɛ = Ke i σ = s/r Case A β + σ β σ > ɛ Case B β + σ β σ < ɛ

19 A SINISTER STRANGER TURNS UP IN THE VILLAGE Dextral Satsuma snail Sinistral Satsuma snail

20 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL Population Distribution Map 1) Regions with only Dextral Satsuma 2) Regions with only Sinistral Satsuma 3) Regions with Dextral and Sinistral Satsuma 4) Regions with Dextral/ Sinistral Satsuma and Pareas iwasakii

21 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL But the dextral snail-eating specialist how good is with the sinistral snails? Success rate 100% Success rate 12.5% Pareas iwasakii snake attacks to dextral and sinistral Satsuma snails are displayed in the attached videos or in the link

22 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL HYPOTHESIS OF THE MODEL. 1) The sinistral/dextral satsuma snails can be considered as two different populations with the same intrinsic biological parameters growth rate r and carrying capacity K). Copulation between them is usually strongly impeded by genital and behavioural mismatches. They are in mutual competition for the resources. The dextral snail population is consolidated while sinistral snails appear as a mutant population. 2) The Pareas iwasakii snake is a dextral snail-eating specialist predator due to adaptation to the surrounding environment. It faces difficulties in hunting sinistral snails. The dextral and sinistral snail hunting efficiencies are approximately e 1 = 1 and e 2 = There exist other pareatic snakes with a less degree of asymmetry. 3) The three populations form a closed homogenous ecosystem.

23 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL X 1 t): Dextral Satsuma population X 2 t): Sinistral Satsuma population Yt): Pareas iwasakii population dx 1 = rx 1 1 X 1 + X ) 2 K δx 1 Y dx 2 = rx 2 1 X 1 + X ) 2 K δx 2 Y dy = sy + βδx 1 + δx 2 )Y

24 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL X 1 t): Dextral Satsuma population X 2 t): Sinistral Satsuma population Yt): Pareas iwasakii population dx 1 = rx 1 1 X 1 + X ) 2 K δx 1 Y dx 2 = rx 2 1 X 1 + X ) 2 K δx 2 Y dy = sy + βδx 1 + δx 2 )Y

25 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL X 1 t): Dextral Satsuma population X 2 t): Sinistral Satsuma population Yt): Pareas iwasakii population dx 1 = rx 1 1 X 1 + X ) 2 K δx 1 Y dx 2 = rx 2 1 X 1 + X ) 2 K δx 2 Y dy = sy + βδx 1 + δx 2 )Y

26 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL X 1 t): Dextral Satsuma population X 2 t): Sinistral Satsuma population Yt): Pareas iwasakii population Holling hypothesis Time devoted to searching, hunting and handling preys is fixed T = T S + T H cte

27 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL X 1 t): Dextral Satsuma population X 2 t): Sinistral Satsuma population Yt): Pareas iwasakii population Holling hypothesis Time devoted to searching, hunting and handling preys is fixed T = T S + T H cte δx 1 = AT S X 1 e 1 δx 2 = AT S X 2 e 2 δx 1 e 1 X 1 = δx 2 e 2 X 2 δx i hunted snails A Area supervised by unit time e i Hunting eficiency on prey i

28 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL X 1 t): Dextral Satsuma population X 2 t): Sinistral Satsuma population Yt): Pareas iwasakii population Holling hypothesis Time devoted to searching, hunting and handling preys is fixed T = T S + T H cte δx 1 = AT S X 1 e 1 δx 2 = AT S X 2 e 2 δx 1 e 1 X 1 = δx 2 e 2 X 2 δx i = T H = t h δx 1 + δx 2 ) e i AT X i 1 + t h Ae 1 X 1 + e 2 X 2 ) δx i hunted snails A Area supervised by unit time e i Hunting eficiency on prey i t h handling time per captured prey unit cte

29 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL X 1 t): Dextral Satsuma population X 2 t): Sinistral Satsuma population Yt): Pareas iwasakii population Holling hypothesis Time devoted to searching, hunting and handling preys is fixed T = T S + T H cte δx 1 = AT S X 1 e 1 δx 2 = AT S X 2 e 2 δx 1 e 1 X 1 = δx 2 e 2 X 2 δx i = T H = t h δx 1 + δx 2 ) e i AT X i 1 + t h Ae 1 X 1 + e 2 X 2 ) δx i hunted snails A Area supervised by unit time e i Hunting eficiency on prey i t h handling time per captured prey unit cte

30 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL X 1 t): Dextral Satsuma population X 2 t): Sinistral Satsuma population Yt): Pareas iwasakii population dx 1 = rx 1 1 X 1 + X ) 2 K δx 1 Y dx 2 = rx 2 1 X 1 + X ) 2 K δx 2 Y dy = sy + βδx 1 + δx 2 )Y

31 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL X 1 t): Dextral Satsuma population X 2 t): Sinistral Satsuma population Yt): Pareas iwasakii population dx 1 = rx 1 1 X 1 + X ) 2 K ae 1 ATX 1 Y 1 + t h Ae 1 X 1 + e 2 X 2 ) dx 2 = rx 2 1 X 1 + X ) 2 K ae 2 ATX 2 Y 1 + t h Ae 1 X 1 + e 2 X 2 ) dy = sy + baate 1X 1 + e 2 X 2 )Y 1 + t h Ae 1 X 1 + e 2 X 2 )

32 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL X 1 t): Dextral Satsuma population X 2 t): Sinistral Satsuma population Yt): Pareas iwasakii population dx 1 = rx 1 1 X 1 + X ) 2 K ae 1 ATX 1 Y 1 + t h Ae 1 X 1 + e 2 X 2 ) dx 2 = rx 2 1 X 1 + X ) 2 K ae 2 ATX 2 Y 1 + t h Ae 1 X 1 + e 2 X 2 ) dy = sy + baate 1X 1 + e 2 X 2 )Y 1 + t h Ae 1 X 1 + e 2 X 2 )

33 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL X 1 t): Dextral Satsuma population X 2 t): Sinistral Satsuma population Yt): Pareas iwasakii population dx 1 = rx 1 1 X 1 + X ) 2 K ae 1 ATX 1 Y 1 + t h Ae 1 X 1 + e 2 X 2 ) dx 2 = rx 2 1 X 1 + X ) 2 K ae 2 ATX 2 Y 1 + t h Ae 1 X 1 + e 2 X 2 ) dy = sy + baate 1X 1 + e 2 X 2 )Y 1 + t h Ae 1 X 1 + e 2 X 2 ) Introducing non-dimensional variables: τ = rt, x i = Xi K, and non-dimensional coefficients: ɛ i = t hake i, σ = s r, y = ae1at Y, r β = bt t h, α = e2 e 1

34 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL x 1 τ): Non-dimensional dextral Satsuma population x 2 τ): Non-dimensional sinistral Satsuma population yτ): Non-dimensional Pareas iwasakii population dx 1 dτ dx 2 dτ dy dτ P 0 0, 0, 0) P 1 P 2 0, = x 1 1 x 1 x 2 ) = x 2 1 x 1 x 2 ) = β σ)y σ, 0, β[βɛ 11+ɛ 1 )σ] ɛ 1 βσ) ɛ 1 βσ) 2 ) ) σ, β[βɛ 21+ɛ 1 )σ] ɛ 2 βσ) αɛ 2 βσ) 2 R 12 = µ, 1 µ, 0) Stationary points x 1 y 1 + ɛ 1 x 1 + αx 2 ) αx 2 y 1 + ɛ 1 x 1 + αx 2 ) βy 1 + ɛ 1 x 1 + αx 2 )

35 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL

36 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL x 1 τ): Non-dimensional dextral Satsuma population x 2 τ): Non-dimensional sinistral Satsuma population yτ): Non-dimensional Pareas iwasakii population dx 1 dτ dx 2 dτ dy dτ = x 1 1 x 1 x 2 ) = x 2 1 x 1 x 2 ) = β σ)y x 1 y 1 + ɛ 1 x 1 + αx 2 ) αx 2 y 1 + ɛ 1 x 1 + αx 2 ) βy 1 + ɛ 1 x 1 + αx 2 )

37 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL x 1 τ): Non-dimensional dextral Satsuma population x 2 τ): Non-dimensional sinistral Satsuma population yτ): Non-dimensional Pareas iwasakii population dρ dτ dθ dτ dy dτ = ρ[1 ρcos θ + sin θ)] ρy[1 1 α) sin2 θ] 1 + ρɛ 1 cos θ + α sin θ) 1 α)ρ sin θ cos θy = 1 + ρɛ 1 cos θ + α sin θ) βy = β σ)y 1 + ρɛ 1 cos θ + α sin θ) ρτ) = x 1τ) 2 + x 2τ) 2 : Non-dimensional Satsuma population measure θτ) = arctan x 2τ) x 1 : Sinistral/Dextral Satsuma population ratio measure τ) yτ): Non-dimensional Pareas iwasakii population

38 A HOLLING TYPE II PREDATOR-TWO PREY VARIANT MODEL x 1 τ): Non-dimensional dextral Satsuma population x 2 τ): Non-dimensional sinistral Satsuma population yτ): Non-dimensional Pareas iwasakii population dρ dτ dθ dτ dy dτ = ρ[1 ρcos θ + sin θ)] ρy[1 1 α) sin2 θ] 1 + ρɛ 1 cos θ + α sin θ) 1 α)ρ sin θ cos θy = 1 + ρɛ 1 cos θ + α sin θ) > 0 βy = β σ)y 1 + ρɛ 1 cos θ + α sin θ) ρτ) = x 1τ) 2 + x 2τ) 2 : Non-dimensional Satsuma population measure θτ) = arctan x 2τ) x 1 : Sinistral/Dextral Satsuma population ratio measure τ) yτ): Non-dimensional Pareas iwasakii population dθ dτ > 0: The dextral Satsuma snail population is replaced by the sinistral variant

39 LIMIT CYCLE TO LIMIT CYCLE ORBIT Final snapshot of Video 1

40 LIMIT CYCLE TO STATIONARY POINT ORBIT Final snapshot of Video 2

41 LIMIT CYCLE TO DEXTRAL SATSUMA AND PAREAS EXTINCTION Final snapshot of Video 3

42 BIBLIOGRAPHY 1. M. Hoso, T. Asami, M. Hori, Right-handed snakes: convergent evolution of asymmetry for functional specialization, Biol. Lett ) M. Hoso, Y. Kameda, S.P. Wu, T. Asami, M. Kato, M. Hori, A speciation gene for left-right reversal in snails results in anti predator adaptation, Nature Commun ) M. Hoso, Non-adaptive speciation of snails by left-right reversal is facilitated on oceanic islands, Contr. Zool. 812) 2012) E. Gittenberger, T.D. Hamann, T. Asami, Chiral Speciation in Terrestrial Pulmonate Snails, PLoS ONE 74) 2012) e D. Mukherjee, The effect of refuge and inmigration in a predator-prey system in the presence of a competitor for the prey, Nonlinear Analysis: RWA ) P. Danaisawadi, T. Asami, H. Ota, Ch. Sutcharit, S. Panha, Subtle asymmetries in the snail-eating snake Pareas carinatus Reptilia: Pareatidae), J. Ethol ) G. Seo, D.L. DeAngelis, A Predator-Prey Model with a Holling Type I Functional Response Including a Predator Mutual Interference, J. Nonlinear Sci ) J. Hofbauer, K. Sigmund, On the stabilizing effect of predators and competitors on ecological communities, J. Math. Biol ) J. Sugie, R. Kohno, R. Miyazaki, On a Predator-Prey System of Holling Type, Proc. Amer. Math. Soc. 1257) 1997) A. Gasull, A. Guillamon, Non-existence of limit cycles for some predator-prey systems, Proceedings of Equadiff91, pp World Sci A. Gasull, A. Guillamon, Non-existence, uniqueness of limit cycles and center problem in a system that includes predator-prey systems and generalized Lienard equations, Differ. Equ. Dyn. Syst. 34) 1995)

43 END OF THE PRESENTATION Thanks for your attention