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- 1. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Eﬀect of the mortality on a density-dependent model with a predator-prey relationship T. Mtar1 R. Fekih-Salem2 T. Sari3 1,2LAMSIN-ENIT University of Tunis El-Manar, TUNISIA 3UMR itap, INRAE University SupAgro, Montpellier, FRANCE CARI’2020 14-17 Octobre, 2020
- 2. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Table of Contents 1 Mathematical model 2 Existence of steady states 3 Local stability of steady states 4 Numerical simulations 5 Conclusion
- 3. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Table of Contents 1 Mathematical model 2 Existence of steady states 3 Local stability of steady states 4 Numerical simulations 5 Conclusion
- 4. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Mathematical model ˙S = D (Sin − S) − f1 (S, x2) x1 − f2 (S, x1) x2, ˙x1 = (f1 (S, x2) − D1) x1, ˙x2 = (f2 (S, x1) − D2) x2, (1) Di = αi D + ai , i = 1, 2 M. El Hajji. . How can inter-speciﬁc interferences explain coexistence or conﬁrm the competitive exclusion principle in a chemostat ?, International Journal of Biomathematics, 11, 2018. T. Mtar, R. Fekih-Salem, T. Sari. . Interspeciﬁc density-dependent model of predator-prey relationship in the chemostat, International Journal of Biomathematics (2020).
- 5. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Assumptions on the model For i = 1, 2, j = 1, 2, i = j, we assume that (H0): fi : R2 + −→ R+ is continuously diﬀerentiable (H1): fi (0, xj ) = 0, for all xj ≥ 0 (H2): ∂fi ∂S (S, xj ) > 0, for all S ≥ 0, x1 > 0 and x2 ≥ 0 (H3): ∂f1 ∂x2 (S, x2) < 0 and ∂f2 ∂x1 (S, x1) > 0, for all S > 0, x1 ≥ 0 and x2 ≥ 0 (H4): f2(S, 0) = 0, for all S > 0
- 6. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Assumptions on the model For i = 1, 2, j = 1, 2, i = j, we assume that (H0): fi : R2 + −→ R+ is continuously diﬀerentiable (H1): fi (0, xj ) = 0, for all xj ≥ 0 (H2): ∂fi ∂S (S, xj ) > 0, for all S ≥ 0, x1 > 0 and x2 ≥ 0 (H3): ∂f1 ∂x2 (S, x2) < 0 and ∂f2 ∂x1 (S, x1) > 0, for all S > 0, x1 ≥ 0 and x2 ≥ 0 (H4): f2(S, 0) = 0, for all S > 0 Proposition For any non-negative initial conditions, all solutions of system (1) are bounded and remain non-negative for all t > 0. In addition, the set Ω = (S, x1, x2) ∈ R3 +; S + x1 + x2 ≤ Sin is positively invariant and is a global attractor for the dynamics (1)
- 7. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Table of Contents 1 Mathematical model 2 Existence of steady states 3 Local stability of steady states 4 Numerical simulations 5 Conclusion
- 8. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Existence of steady states 0 = D (Sin − S) − f1 (S, x2) x1 − f2 (S, x1) x2, 0 = (f1 (S, x2) − D1) x1, 0 = (f2 (S, x1) − D2) x2, (2)
- 9. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Existence of steady states 0 = D (Sin − S) − f1 (S, x2) x1 − f2 (S, x1) x2, 0 = (f1 (S, x2) − D1) x1, 0 = (f2 (S, x1) − D2) x2, (2) S = Sin − D1x1/D − D2x2/D (3)
- 10. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Existence of steady states 0 = D (Sin − S) − f1 (S, x2) x1 − f2 (S, x1) x2, 0 = (f1 (S, x2) − D1) x1, 0 = (f2 (S, x1) − D2) x2, (2) S = Sin − D1x1/D − D2x2/D (3) f1(Sin − D1x1/D − D2x2/D, x2) = D1 ⇒ F1(x1) = x2 f2(Sin − D1x1/D − D2x2/D, x1) = D2 ⇒ F2(x1) = x2
- 11. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion δ deﬁned by the equation Sin = D1x1/D + D2x2/D Figure: Deﬁnition of function F1 Figure: Deﬁnition of function F2
- 12. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Proposition Under the assumptions (H0)-(H4), system (1) has the following steady states: E0 = (Sin, 0, 0), that always exists, E1 = (Sin − D1 ˜x1/D, ˜x1, 0), where ˜x1 is the unique solution of f1(Sin − D1x1/D, 0) = D1 (4) It exists if and only if f1(Sin, 0) > D1, E∗ = (S∗, x∗ 1 , x∗ 2 ), where S∗ is given by S∗ = Sin − D1x∗ 1 /D − D2x∗ 2 /D (5) and (x∗ 1 , x∗ 2 ) are the solution of F1(x1) = x2 and F2(x1) = x2 (6)
- 13. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Multiplicity of positive steady states Proposition • If ˜x1 < x1 1 < x2 1 , then there is no positive steady state • If x1 1 < ˜x1 < x2 1 , then there exists at least one positive steady state. Generically, the system has an odd number of positive steady states • If x1 1 < x2 1 < ˜x1. then generically system (1) has no positive steady state or an even number of positive steady states
- 14. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Multiplicity of positive steady states Figure: (a) 0 intersection Figure: (b) Odd number of intersection Figure: (c) Even number of intersection
- 15. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Table of Contents 1 Mathematical model 2 Existence of steady states 3 Local stability of steady states 4 Numerical simulations 5 Conclusion
- 16. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Local stability of steady states Let J the Jacobian matrix of (1) at (S, x1, x2): J = −D − x1E − x2F −f1(S, x2) − x2H x1G − f2(S, x1) x1E f1(S, x2) − D1 −x1G x2F x2H f2(S, x1) − D2 where E = ∂f1 ∂S , F = ∂f2 ∂S , G = − ∂f1 ∂x2 , H = ∂f2 ∂x1 . (7)
- 17. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Local stability of steady states Let J the Jacobian matrix of (1) at (S, x1, x2): J = −D − x1E − x2F −f1(S, x2) − x2H x1G − f2(S, x1) x1E f1(S, x2) − D1 −x1G x2F x2H f2(S, x1) − D2 where E = ∂f1 ∂S , F = ∂f2 ∂S , G = − ∂f1 ∂x2 , H = ∂f2 ∂x1 . (7) Proposition E0 is LES if and only if f1(Sin, 0) < D1 E1 is LES if and only if f2 (Sin − D1 ˜x1/D, ˜x1) < D2
- 18. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Local stability of steady states The Jacobian matrix at E∗ = (S∗, x∗ 1 , x∗ 2 ) is given by J∗ = −D − x∗ 1 E − x∗ 2 F −D1 − x∗ 2 H x∗ 1 G − D2 x∗ 1 E 0 −x∗ 1 G x∗ 2 F x∗ 2 H 0 The characteristic polynomial is given by: P(λ) = λ3 + c1λ2 + c2λ + c3, where c1 = D + Ex∗ 1 + Fx∗ 2 . c2 = D1Ex∗ 1 + D2Fx∗ 2 + x∗ 1 x∗ 2 (GH + EH − FG). c3 = x∗ 1 x∗ 2 (DGH + D2EH − D1FG).
- 19. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Routh-Hurwitz criterion Since c1 > 0, according to the Routh-Hurwitz criterion, E∗ is LES if and only if c3 > 0 and c4 = c1c2 − c3 > 0 (8)
- 20. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Routh-Hurwitz criterion Since c1 > 0, according to the Routh-Hurwitz criterion, E∗ is LES if and only if c3 > 0 and c4 = c1c2 − c3 > 0 (8) Lemma One has c3 = x∗ 1 x∗ 2 (F2(x∗ 1 ) − F1(x∗ 1 ))D2F (D2E/D + G)
- 21. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Table of Contents 1 Mathematical model 2 Existence of steady states 3 Local stability of steady states 4 Numerical simulations 5 Conclusion
- 22. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Numerical simulations We assumed that the growth functions are given by: f1(S, x2) = m1S K1 + S 1 1 + x2 Ki , f2(S, x1) = m2S K2 + S x1 L2 + x1 The parameter values using for numerical simulations are chosen as follows: m1 = 4, K1 = 2, Ki = 3 m2 = 8, K2 = 0.2, L2 = 0.1 α1 = α2 = 1, a1 = 0.3, a2 = 0.2, D = 1
- 23. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Hopf bifurcation (D = 1 ﬁxed) Figure: (a) Change of sign of c4 when Sin increases Figure: (b) Variation of a pair of complex-conjugate eigenvalues as Sin increases
- 24. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Occurrence of limit cycle Sin 6.27 > Scr in 6.265 Figure: (c) Limit cycle when the oscillations are sustained
- 25. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Table of Contents 1 Mathematical model 2 Existence of steady states 3 Local stability of steady states 4 Numerical simulations 5 Conclusion
- 26. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Conclusion We have considered a density-dependent model of two species competing on one resource with a predator-prey relationship and distinct removal rates.
- 27. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Conclusion We have considered a density-dependent model of two species competing on one resource with a predator-prey relationship and distinct removal rates. We prove that the system can have three types of steady states.
- 28. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Conclusion We have considered a density-dependent model of two species competing on one resource with a predator-prey relationship and distinct removal rates. We prove that the system can have three types of steady states. We give the conditions for their existence and stability
- 29. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Conclusion We have considered a density-dependent model of two species competing on one resource with a predator-prey relationship and distinct removal rates. We prove that the system can have three types of steady states. We give the conditions for their existence and stability We show that the system can have a multiplicity of coexistence steady states
- 30. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Conclusion We have considered a density-dependent model of two species competing on one resource with a predator-prey relationship and distinct removal rates. We prove that the system can have three types of steady states. We give the conditions for their existence and stability We show that the system can have a multiplicity of coexistence steady states We succeeded in ﬁnding a set of parameters such that the positive steady state loses its stability through a supercritical Hopf bifurcation
- 31. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion Conclusion We have considered a density-dependent model of two species competing on one resource with a predator-prey relationship and distinct removal rates. We prove that the system can have three types of steady states. We give the conditions for their existence and stability We show that the system can have a multiplicity of coexistence steady states We succeeded in ﬁnding a set of parameters such that the positive steady state loses its stability through a supercritical Hopf bifurcation We show through numerical simulations the appearance of a stable limit cycle
- 32. Predator-prey relationship T. Mtar & R. Fekih-Salem & T. Sari Mathematical model Existence of steady states Local stability of steady states Numerical simulations Conclusion