Competitive ecosystems: continuous and discrete models

- vironment. We assume that on two given patch, the rice’s density increases in the logistic form and decreases by BPH’s eating, while the BPH’s density increase by eating rice. The BPHs will be extinct when there is no rice on two patches. We further assume that BPH individuals can move between the two patches and this dispersal process acts on a fast time-scale of the demography, the competition and predation processes act on a slow time-scale on the two local patches. Denote ni is the densities of rice respectively on patch i, i ∈ {1, 2}. piA, piJ are the densities of BPH in mature stage and in egg stage respectively on patch i, i ∈ {1, 2}. Parameters r1, r2 and K are the growth rate and the carrying capacity of rice on each rice field, respectively. Parameters diA, diJ are the natural death rates of BPH in mature stage and BPH in egg stage respectively on patch i, i ∈ {1, 2}. And ai represents the eating rates of BPH, ei is the parameter related to BPH recruitment as a consequence of prey-predator interaction, respectively on patch i, i ∈ {1, 2}. We suppose that m, m are the dispersal rates of BPH in mature stage from region 1 to region 2 and opposite. Parameter ε represents the ratio between the two time-scales t = ετ . Then the completed model is given as follows: dn1 dτ = ε [ r1n1 ( 1− n1 K ) − a1n1p1A ] , dn2 dτ = ε [ r2n2 ( 1− n2 K ) − a2n2p2A ] , dp1A dτ = ε (− d1Ap1A + α1p1J)+ (mp2A −mp1A), dp2A dτ = ε (− d2Ap2A + α2p2J)+ (mp1A −mp2A), dp1J dτ = ε (− d1Jp1J − α1p1J + e1a1n1p1A), dp2J dτ = ε (− d2Jp2J − α2p2J + e2a2n2p2A), (4.12) For biological reasons, we only consider the solutions (n1, n2, p1A, p2A, p1J , p2J) with non-negative initial values n1(0) ≥ 0, n2(0) ≥ 0, p1A(0) ≥ 0, p2A(0) ≥ 0, p1J(0) ≥ 0, p2J(0) ≥ 0. 78 Figure 4.9: Compare the density of rice on patch 1 between the original model and the reduced one. The case: rice wins globally in competition. Parameters values are chosen as follows: r1 = 0.7; r2 = 0.2; K = 40; a1 = 0.2; a2 = 0.2; e1 = 0.05; e2 = 0.05; α1 = 0.2; α2 = 0.3; m = 0.3; m = 0.7; d1A = 0.3; d2A = 0.5; d1J = 0.2; d2J = 0.3. 4.2.3 Analysis and Discussion The well-posedness of model (4.12) In this section, we prove the well-posedness of the model (4.12) by analyzing some properties of its solutions through following theorem: Theorem 4.2.1. The system (4.12) has a unique solution with respect to the initial condition (n1(0), n2(0), p1A(0), p2A(0), p1J(0), p2J(0)) ∈ R6+. Moreover, the solutions of (4.12) have the non-negativity property and are bounded on R4+. The proof for Theorem 4.2.1 is similar to the one of Theorem 2.3.1. In the next section, we perform a model reduction as a means of further inves- tigating of the complete model. Reduced Model Model 4.12 is written by coupling two different processes that act at different time scales so it fits in the general form (1.5) presented in Chapter 1. We now proceed to transform it into the so-called slow-fast form so that we get the associated reduced system that serves to study some aspects of its asymptotic behavior. To transform system 4.12 into slowfast form, the key point is making emerge its slow variables. 79 The natural choice is the total densities of of BPH in mature stage over patches because these variables are kept constant through dispersals, the fast dynamics, and so they evolve at the slow time scale, the one that competition process acts at. The slow variables are therefore n1, n2, p1J , p2J and pA = p1A + p2A. The change of variables, using the notation of Chapter 1, that transforms the system (4.12) into slow-fast form is the following: z ∈ R6 is transformed into z = (x, y) ∈ R×R5 where x = (p1A) and y = (n1, n2, p1J , p2J , pA). The resulting system reads  dp1A dτ = [ m(pA − p1A)−mp1A ] + ε (− d1Ap1A + α1p1J), dpA dτ = ε [− d2A(pA − p1A) + α2p2J − d1Ap1A + α1p1J], dn1 dτ = ε [ r1n1 ( 1− n1 K ) − a1n1p1A ] , dn2 dτ = ε [ r2n2 ( 1− n2 K ) − a2n2p2A ] , dp1J dτ = ε (− d1Jp1J − α1p1J + e1a1n1p1A), dp2J dτ = ε (− d2Jp2J − α2p2J + e2a2n2p2A) (4.13) which has the standard slow-fast form (1.5) presented in Chapter 1. dx dτ = F (x, y) + S(x, y), dy dτ = G(x, y). (4.14) The reduction process now consists in taking → 0 in the first equation the slowfast form (4.14), dx dτ = F (x, y), and assuming, for constant y, that there exists asymptoti- cally stable equilibrium x∗(y), in building up a reduced system for the slow variables with the following form: dy dt = G(x∗(y), y), (4.15) where t = τ represents the slow time variable. Under the two hypotheses: (H1) the system (4.15) is structurally stable and (H2)  is small enough, the asymptotic behavior of system (4.13) can be studied through system (4.15). Now, we build up the reduced system associated with the system (4.12). First, from the fast part of the slow-fast system (4.12), it implies that the following asymptotically stable 80 equilibrium:  p∗1A = m m+m pA = µ1pA, p∗2A = m m+m pA = µ2pA. (4.16) We substitute this equilibrium into in the slow part of the slow-fast system (4.12), the reduced model reads dn1 dt = n1 [ r1 ( 1− n1 K )− a1µ1pA], dn2 dt = n2 [ r2 ( 1− n2 K )− a2µ2pA], dpA dt = −(d1Aµ1 + d2Aµ2)pA + (α1p1J + α2p2J), dp1J dt = −d1Jp1J − α1p1J + e1a1n1µ1pA, dp2J dt = −d2Jp2J − α2p2J + e2a2n2µ2pA, (4.17) where µ1 = m m+m and µ2 = m m+m . In the next subsection, we will show conditions for global/local stabilities of the equilibria of the reduced model ensuring the global/local structural stabilities, i.e. a guarantee for Hypothesis H1. Hypothesis H2 is met because we assume  is very small due to the existence of the time scales (see Chapter 1). Local Analysis of the Reduced Model We first look at equilibria of the reduced model. There are always three equilibria on the axes. They are E0(0, 0, 0, 0, 0), E1(0, K, 0, 0, 0), and E2(K, 0, 0, 0, 0). Equilib- ria on the bounded space areE3(K,K, 0, 0, 0) which always exists, E4(nˆ1, 0, pˆA, pˆ1J , 0) exists under condition 1/K < 1/nˆ1 where nˆ1 = (d1Aµ1 + d2Aµ2)(d1J + α1) µ1e1a1α1 , pˆA = r1 ( 1− nˆ1 K ) a1µ1 , pˆ1J = (d1Aµ1 + d2Aµ2)pˆA α1 , and E5(0, n¯2, p¯A, 0, p¯2J) exists if 1/K < 1/n¯2 where n¯2 = (d1Aµ1 + d2Aµ2)(d2J + α2) µ2e2a2α2 , p¯A = r2 ( 1− n¯2 K ) a2µ2 , p¯2J = (d1Aµ1 + d2Aµ2)p¯A α2 . 81 The last equilibrium is the interior one E6(n ∗ 1, n ∗ 2, p ∗ A, p ∗ 1J , p ∗ 2J) which exists if the following conditions are satisfied: max{ 1 nˆ1 , 1 n¯2 } < 1 K < 1 nˆ1 + 1 n¯2 , where p∗A = ( α1e1a1µ1 d1J + α1 + α2e2a2µ2 d2J + α2 ) − d1Aµ1 + d2Aµ2 K a1µ1 r × α1e1a1µ1 d1J + α1 + a2µ2 r2 × α2e2a2µ2 d2J + α2 , n∗1 = K ( 1− a1µ1 r1 p∗A ) , n∗2 = K ( 1− a2µ2 r1 p∗A ) , p∗1J = e1a1µ1 (d1J + α1) n∗1p ∗ A, p ∗ 2J = e2a2µ2 (d2J + α2) n∗2p ∗ A. We study equilibria and stability of (4.17) via the Lyapunov’s first method (1.4.5) and the Jacobian matrix J(n1, n2, pA, p1J , p2J) of the reduced model is given by J =  − r1 K n1 0 −a1µ1n1 0 0 0 − r2 K n2 −a2µ2n2 0 0 0 0 −(d1Aµ1 + d2Aµ2) α1 α2 e1a1µ1pA 0 e1a1µ1n1 −(d1J + α1) 0 0 e2a2µ2pA e2a2µ2n2 0 −(d2J + α2)  . To present briefly, we put d = d1Aµ1 + d2Aµ2. At the point E0(0, 0, 0, 0, 0), the Jacobian matrix reads as follows: J(E0) =  r1 0 0 0 0 0 r2 0 0 0 0 0 −d α1 α2 0 0 0 −(d1J + α1) 0 0 0 0 0 −(d2J + α2)  . The matrix has two positive real eigenvalues λ1 = r1 > 0, λ2 = r2 > 0 and thus (0, 0, 0, 0, 0) is always unstable. Similar to the point E0, the Jacobian matrix at the point E1(0, K, 0, 0, 0) is J(E1) =  r1 0 0 0 0 0 r2 −a2µ2K 0 0 0 0 −d α1 α2 0 0 0 −(d1J + α1) 0 0 0 e2a2µ2K 0 −(d2J + α2)  . The matrix has one eigenvalue λ1 = r > 0 and thus (0, K, 0, 0, 0) is always unstable. We also have the same result for the point E2(K, 0, 0, 0, 0), E2 is always unstable. 82 At the point E3(K,K, 0, 0, 0), a straightforward calculation leads to the following Jacobian matrix J(E3) =  −r1 0 −a1µ1K 0 0 0 −r2 −a2µ2K 0 0 0 0 −d α1 α2 0 0 e1a1µ1K −(d1J + α1) 0 0 0 e2a2µ2K 0 −(d2J + α2)  . det(J−λI) = ∣∣∣∣∣∣∣∣∣∣∣∣∣ −r1 − λ 0 −a1µ1K 0 0 0 −r2 − λ −a2µ2K 0 0 0 0 −d− λ α1 α2 0 0 e1a1µ1K −(d1J + α1)− λ 0 0 0 e2a2µ2K 0 −(d2J + α2)− λ ∣∣∣∣∣∣∣∣∣∣∣∣∣ . = (−r1 − λ)(−r2 − λ)M, where M = ∣∣∣∣∣∣∣∣ −(d1Aµ1 + d2Aµ2)− λ α1 α2 e1a1µ1K −(d1J + α1)− λ 0 e2a2µ2K 0 −(d2J + α2)− λ ∣∣∣∣∣∣∣∣ = λ3 + P1λ 2 + P2λ+ P3, P1 =d1Aµ1 + d2Aµ2 + d1J + α1 + d2J + α2, P2 =(d1Aµ1 + d2Aµ2)(d1J + α1 + d2J + α2) + (d1J + α1)(d2J + α2) − (α1e1a1µ1K + α2e2a2µ2K), P3 =(d1Aµ1 + d2Aµ2)(d1J + α1)(d2J + α2)− α1e1a1µ1K(d2J + α2) − α2e2a2µ2K(d1J + α1). The Routh-Hurwitz criteria for stable stability of the equilibrium E3 are: P1 > 0, P2 > 0, P1P2 − P3 > 0. To summarize, the point E3(K,K, 0, 0, 0) is stable if the following condition is satisfied: (d1Aµ1 + d2Aµ2)(d1J +α1)(d2J +α2) > α1e1a1µ1K(d2J +α2) +α2e2a2µ2K(d1J +α1). At the point E4 = (nˆ1, 0, pˆA, pˆ1J , 0), we have J(E4) =  − r1 K nˆ1 0 −a1µ1nˆ1 0 0 0 r2 − a2µ2pˆA 0 0 0 0 0 −d α1 α2 e1a1µ1pA 0 e1a1µ1nˆ1 −(d1J + α1) 0 0 e2a2µ2pA 0 0 −(d2J + α2)  . 83 det(J − λI) = = ∣∣∣∣∣∣∣∣∣∣∣∣∣ − r1 K nˆ1 − λ 0 −a1µ1nˆ1 0 0 0 r2 − a2µ2pˆA − λ 0 0 0 0 0 −d− λ α1 α2 e1a1µ1pA 0 e1a1µ1nˆ1 −(d1J + α1)− λ 0 0 e2a2µ2pA 0 0 −(d2J + α2)− λ ∣∣∣∣∣∣∣∣∣∣∣∣∣ = ( r2 − a2µ2pˆA − λ )(− (d2J + α2)− λ)N. where N = ∣∣∣∣∣∣∣∣ − r1 K nˆ1 − λ −a1µ1nˆ1 0 0 −d− λ α1 e1a1µ1pA e1a1µ1nˆ1 −(d1J + α1)− λ ∣∣∣∣∣∣∣∣ = λ3 +Q1λ 2 +Q2λ+Q3. Q1 = r1 K nˆ1 + d+ d1J + α1. Q2 = d(d1J + α1) + r1 K nˆ1(d+ d1J + α1)− α1e1a1µ1(ˆn)1 = r1 K nˆ1(d+ d1J + α1). Q3 = r1 K nˆ1d(d1J + α1) + α1e1a 2 1µ 2 1nˆ1pˆA − r1 K nˆ1α1e1a1µ1nˆ1 = α1e1a 2 1µ 2 1nˆ1pˆA. The Routh-Hurwitz criteria for stable stability of the equilibrium E4 are: Q1 > 0, Q3 > 0, Q1Q2 − Q3 > 0, λ1 = −a2µ2pˆA + r2 < 0, λ2 = −(d2J + α2) < 0. To summarize, the point E4 = (nˆ1, 0, pˆA, pˆ1J , 0) is stable if the following conditions are satisfied: r1 ( 1− nˆ1 K ) a1µ1 ≥ 0, −a2µ2pˆA + r2 < 0,( r1 K nˆ1 + d+ d1J + α1 )( r1 K nˆ1(d+ d1J + α1) ) > α1e1a 2 1µ 2 1nˆ1pˆA. Similar calculations to the point E5 = (0, n¯2, p¯A, 0, p¯2J), the conditions for its stability are given by r2 ( 1− nˆ2 K ) a2µ2 ≥ 0, −a1µ1pˆA + r1 < 0,( r2 K nˆ2 + d+ d2J + α2 )( r2 K nˆ2(d+ d2J + α2) ) > α2e2a 2 2µ 2 2nˆ2pˆA. 84 At the point E6 = (n ∗ 1, n ∗ 2, p ∗ A, p ∗ 1J , p ∗ 2J), the Jacobian matrix reads J(E6) =  − r1 K n∗1 0 −a1µ1n∗1 0 0 0 − r2 K n∗2 −a2µ2n∗2 0 0 0 0 −d α1 α2 e1a1µ1p ∗ A 0 e1a1µ1n ∗ 1 −(d1J + α1) 0 0 e2a2µ2p ∗ A e2a2µ2n ∗ 2 0 −(d2J + α2)  det(J − λI) = = ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ − r1 K n∗1 − λ 0 −a1µ1n∗1 0 0 0 − r2 K n∗2 − λ −a2µ2n∗2 0 0 0 0 −d− λ α1 α2 e1a1µ1p ∗ A 0 e1a1µ1n ∗ 1 −(d1J + α1)− λ 0 0 e2a2µ2p ∗ A e2a2µ2n ∗ 2 0 −(d2J + α2)− λ ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ . T = ∣∣∣∣∣∣∣∣ − r2 K n∗2 − λ −a2µ2n∗2 0 0 −d− λ α1 0 e1a1µ1n ∗ 1 −(d1J + α1)− λ ∣∣∣∣∣∣∣∣ = ( − r2 K n∗2 − λ )[ (d+ λ)(d1J + α1 + λ)− α1e1a1µ1n∗1 ] . A11 = ∣∣∣∣∣∣∣∣∣∣∣ − r1 K n∗1 − λ 0 −a1µ1n∗1 0 0 − r2 K n∗2 − λ −a2µ2n∗2 0 0 0 −d− λ α1 e1a1µ1p ∗ A 0 e1a1µ1n ∗ 1 −(d1J + α1)− λ ∣∣∣∣∣∣∣∣∣∣∣ = ( − r1 K n∗1 − λ ) T − e1a1µ1p∗A ( r2 K n∗2 + λ )(− a1µ1n∗1)α1. A12 = ∣∣∣∣∣∣∣∣∣∣∣ 0 −a1µ1n∗1 0 0 − r2 K n∗2 − λ −a2µ2n∗2 0 0 0 −d− λ α1 α2 0 e1a1µ1n ∗ 1 −(d1J + α1)− λ 0 ∣∣∣∣∣∣∣∣∣∣∣ = ( r2 K n∗2 + λ) (− a1µ1n∗1)(d1J + α1 + λ)α2. 85 A21 = ∣∣∣∣∣∣∣∣∣∣∣ 0 − r2 K n∗2 − λ −a2µ2n∗2 0 0 0 −d− λ α1 e1a1µ1p ∗ A 0 e1a1µ1n ∗ 1 −(d1J + α1)− λ 0 e2a2µ2p ∗ A e2a2µ2n ∗ 2 0 ∣∣∣∣∣∣∣∣∣∣∣ = e1a1µ1p ∗ A(−α1)e2a2µ2n∗2 ( − r2 K n∗2 − λ+ a2µ2p∗A ) . A22 = ∣∣∣∣∣∣∣∣∣∣∣ − r2 K n∗2 − λ −a2µ2n∗2 0 0 0 −d− λ α1 α2 0 e1a1µ1n ∗ 1 −(d1J + α1)− λ 0 e2a2µ2p ∗ A e2a2µ2n ∗ 2 0 −(d2J + α2)− λ ∣∣∣∣∣∣∣∣∣∣∣ =− (d2J + α2 + λ)T + α2(d1J + α1 + λ)e2a2µ2n∗2 ( − r2 K n∗2 − λ+ a2µ2p∗A ) . Using Dodgson’s condensation method in [96], det(J − λI) can be calculated as follow det(J − λI) = 1 T ∣∣∣∣∣ A11 A12A21 A22 ∣∣∣∣∣ = 1T (A11A22 − A12A21) =− λ5 −B1λ4 −B2λ3 −B3λ2 −B4λ−B5. Denote r1 K n∗1 = O1, r2 K n∗2 = O2, d1J +α1 = P1, d2J +α2 = P2 and d1Aµ1 +d2Aµ2 = Q, we have B1 = Q+O1 +O2 + P1 + P2, B2 = O1O2 + P1P2 +O1P1 +O2P1 +O2P2 +O1P2 +QO1 +QO2 +QP1 +QP2 − α1e1a1µ1n∗1 − α2e2a2µ2n∗2, B3 = P1P1O1 + P1P2O2 +O1O2P1 +O1O2P2 +QO1O2 +QP1P2 +QO1P1 +QO1P2 +QO2P1 +QO2P2 − α1e1a1µ1n∗1(P2 +O1 +O2) − α2e2a2µ2n∗2(P1 +O1 +O2) + α2e2a22µ22n∗2p∗A + α1e1a21µ21n∗1p∗A, B4 = O1O2P1P2 +QP1P2(O1 +O2) +QO1O2(P1 + P2) − α1e1a1µ1n∗1(P2O1 + P2O2 +O1O2)− α2e2a2µ2n∗2(P1O1 + P1O2 +O1O2) + α2e2a 2 2µ 2 2n ∗ 2p ∗ A(O1 + P1) + α1e1a 2 1µ 2 1n ∗ 1p ∗ A(O2 + P2), B5 = QO1O2P1P2 − α1e1a1µ1n∗1O1O2P2 − α2e2a2µ2n∗2O1O2P1 + α2e2a 2 2µ 2 2n ∗ 2p ∗ AO1P + α1e1a 2 1µ 2 1n ∗ 1p ∗ AO2P2. 86 Using Routh-Hurwitz criterion, we have the following conditions: B1 > 0, B5 > 0, B1B2 > B3, B1B2B3 −B23 −B21B4 +B1B5 > 0, (B1B2B3 −B23 −B21B4 +B1B5)(B1B4 −B5)−B5(B1B2 −B3)2 > 0. (4.18) Stability diagram can be categorized depending on 1 K . A simple analysis shows that if 1 K > 1 nˆ1 + 1 n¯2 then the dynamics system has 4 equilibria: E0, E1, E2, E3 and the equilibria E3 is always asymptotically stable. If the equilibria E6 exists then two equilibria E4, E5 are always unstable. And E4, E5 cannot be both stable. A detail stability analysis is fully provided as above but is not easily categorized because there is a lot of parameters. Figure 4.10: Equilibria and local stability analysis of the reduced model. We present in Figure 4.10 an example where ones can see the whole stability picture of the model depending on 1 K while others are fixed. Precisely, the parameters are chosen as r1 = 0.3, r2 = 0.9, a1 = 0.7, a2 = 0.1, e1 = 0.9, e2 = 0.5, α1 = 0.1, α2 = 0.1, m = 0.8, m¯ = 0.2 and 1 K0 is the positive solution of the following equation of variable X r22n¯2X 2 + r2 [ v + (d1Aµ1 + d2Aµ2)(d2J + α2) v ] X − e2α2a2µ2r2 v = 0, where v = d1Aµ1 + d2Aµ2 + d2J +α2. E6(*) means the equilibrium E6 is stable when the condition 4.18 is satisfied. And in the case (**), the result of the simulations shows that the dynamical system has a limit cycle around the equilibrium E5. Global Analysis of the Reduced Model We now show in this subsection some conditions for global stabilities of equilibria of the reduced model. 87 Theorem 4.2.2. Suppose that a1µ1e1Kα1 d1J + α1 + a2µ2e2Kα2 d2J + α2 ≤ d1Aµ1 + d2Aµ2. (4.19) Then E3(K,K, 0, 0, 0) is globally asymptotically stable on (0,+∞)5. Proof. Let us consider the Lyapunov functional: V1(n1, n2, pA, p1J , p2J) = e1α1K d1J + α1 ( n1 K − 1− log n1 K ) + e2α2K d2J + α2 ( n2 K − 1− log n2 K ) + pA + α1 d1J + α1 p1J + α2 d2J + α2 p2J . It is easy to see that V1(n1, n2, pA, p1J , p2J) ≥ 0 and V1(n1, n2, pA, p1J , p2J) = 0 if and only if n1 = K, n2 = K, pA = 0, p1J = 0, p2J = 0. We compute the derivative of V1 along the solutions of (4.17) as follows V˙1 = e1α1K(n1 −K)n˙1 (d1J + α1)n1K + e2α2K(n2 −K)n˙2 (d2J + α1)n2K + p˙A + α1 d1J + α1 p˙1J + α2 d2J + α2 p˙2J =− e1α1(n1 −K) 2 (d1J + α1)K − e2α2(n2 −K) 2 (d2J + α2)K + pA [ a1µ1e1α1K d1J + α1 + a2µ2e2α2K d2J + α2 − (d1Aµ1 + d2Aµ2) ] . It is seen that V˙1 ≤ 0, with equality if and only if n1 = K,n2 = K and either pA = 0 or a1µ1e1Kα1 d1J + α1 + a2µ2e2Kα2 d2J + α2 = d1Aµ1 + d2Aµ2. In both cases, the only invariant subset M˜ within the set M = {(n1, n2, pA, p1J , p2J) : n1 = K,n2 = K} is M˜ = {(K,K, 0, 0, 0)}. Using LaSalle’s invariance principle (see [61]), we obtained {(K,K, 0, 0, 0)} is asymptotically stable on (0,+∞)5. Remark 4.2.3. The condition for locally asymptotically stability of E3(K,K, 0, 0, 0) is the same as the one for its globally asymptotically stability. Theorem 4.2.4. Suppose that pA > pˆA > r2 a2µ2 , and n1 > K 2 where n1, pA are such that n1 ≤ lim inf t→∞ n1(t) and pA ≤ lim inf t→∞ pA(t). Then E4(nˆ1, 0, pˆA, pˆ1J , 0) is globally asymptotically stable on (0,+∞)5. Proof. Let us consider the Lyapunov functional: V2(n1, n2, pA, p1J , p2J) = hn1 ( n1 nˆ1 − 1− log n1 nˆ1 ) + hn2n2 + hpA ( pA pˆA − 1− log pA pˆA ) + hp1J ( p1J pˆ1J − 1− log p1J ˆp1J ) + hp2Jp2J , 88 with  hn1 = 1 > 0, hn2 = α2e2a2µ2pˆA(d1J + α1) e1α1(d2J + α2)nˆ1(a2µ2pˆA − r2) > 0, hpA = (d1J + α1)pˆA e1α1nˆ1 > 0, hp1J = pˆ1J e1nˆ1 > 0, hp2J = α2(d1J + α1) (d2J + α2)e1α1nˆ1 > 0. It is easily seen that V2(n1, n2, pA, p1J , p2J) ≥ 0 and V2(n1, n2, pA, p1J , p2J) = 0 if and only if n1 = nˆ1, n2 = 0, pA = pˆA, p1J = ˆp1J , p2J = 0. We compute the derivative of V2 along the solutions of (4.17) as follows V˙2 = r1(n1 − nˆ1)2(K − n1 − nˆ1) Kn1nˆ1 − r2hn2n 2 2 K + h2Ja2µ2e2 ( 1− a2µ2pˆA a2µ2pˆA − r2 ) n2(pA − pˆA) − [ p1J(d1J + α1)pˆA pAe1nˆ1 + pˆ1Ja1µ1n1pA nˆ1p1J + r1(K − nˆ1)nˆ1 Kn1 − 3r1 ( 1− nˆ1 K )] − α2(d2J + α1)pˆAp2J e1α1nˆ1pA ≤ 0. Since n1 > K 2 , it is seen that there is t0 ≥ 0 such that n1(t) > K2 for all t ≥ t0 and also that nˆ1 > K 2 . It follows that r1(n1 − nˆ1)2(K − n1 − nˆ1) Kn1nˆ1 ≤ 0. With the condition pA > pˆA > r2 a2µ2 , we have h2Ja2µ2e2 ( 1− a2µ2pˆA a2µ2pˆA − r2 ) n2(pA − pˆA) ≤ 0. Using Cauchy Schwarz inequality, we obtain p1J(d1J + α1)pˆA pAe1nˆ1 + pˆ1Ja1µ1n1pA nˆ1p1J + r2(K − nˆ1)nˆ1 Kn1 − 3r1 ( 1− nˆ1 K ) ≥ 0 with equality if and only if pA pˆA = p1J ˆp1J . This implies that V˙2 ≤ 0, with equality if and only if n1 = nˆ1, n2 = 0, pA pˆA = p1J ˆp1J , p2J = 0. The invariant subsets M˜ within the set M = {(n1, n2, pA, p1J , p2J) : n1 = nˆ1, n2 = 0, pA pˆA = p1J ˆp1J , p2J = 0} is M˜ = {(nˆ1, 0, pˆA, pˆ1J , 0)}. Using Lasalle’s invariance 89 principle (1.4.7), we obtained (nˆ1, 0, pˆA, pˆ1J , 0) is global asymptotically stable on (0,+∞)5. Theorem 4.2.5. Suppose that n1 > K 2 and n2 > K 2 where n1, n2 are such that n1 ≤ lim inf t→∞ n1(t) and n2 ≤ lim inf t→∞ n2(t). Then E6(n ∗ 1, n ∗ 2, p ∗ A, p ∗ 1J , p ∗ 2J) is globally asymptotically stable on (0,+∞)5. Proof. Let us consider the Lyapunov functional: V3(n1, n2, pA, p1J , p2J) = hn1 ( n1 n∗1 − 1− log n1 n∗1 ) + hn2 ( n2 n∗2 − 1− log n2 n∗2 ) + hpA ( pA p∗A − 1− log pA p∗A ) + hp1J ( p1J p∗1J − 1− log p1J p∗1J ) + hp2J ( p2J p∗2J − 1− log p2J p∗2J ) with  h1 = e1α1n ∗ 1 (d1J + α1)p∗A > 0, h2 = e2α2n ∗ 2 (d2J + α2)p∗A > 0, hA = 1 > 0, h1J = α1p ∗ 1J (d1J + α1)p∗A > 0, h2J = α2p ∗ 2J (d2J + α2)p∗A > 0. It is easy to see that V3(n1, n2, pA, p1J , p2J) ≥ 0 and V3(n1, n2, pA, p1J , p2J) = 0 if and only if n1 = n ∗ 1, n2 = n ∗ 2, pA = p ∗ A, p1J = p ∗ 1J , p2J = p ∗ 2J . We compute the derivative of V3 along the solutions of (4.17) V˙3 = h1 [( n1 n∗1 −1 )( r1 ( 1−n1 K ) −a1µ1pA )] +h2 [( n2 n∗2 −1 )( r2 ( 1−n2 K ) −a2µ2pA )] + hA ( 1 p∗A − 1 pA )( − (d1Aµ1 + d2Aµ2)pA + α1p1J + α2p2J ) + h1J ( 1 p∗1J − 1 p1J )( − d1J + α1)p1J + e1a1µ1n1pA ) + h2J ( 1 p∗2J − 1 p2J )( − d2J + α2)p2J + e2a2µ2n2pA ) = e1α1n ∗ 1 (d1J + α1)p∗A [ r1(n− n∗1)(K − n1) Kn∗1 + r1(K − n∗1)n∗1 Kn1 − r1(K − n ∗ 1 K ] + e2α2n ∗ 2 (d2J + α2)p∗A [ r2(n− n∗2)(K − n2) Kn∗2 + r2(K − n∗2)n∗2 Kn2 − r2(K − n ∗ 2 K ] − [ α1p1J pA + α1e1a1µ1n1p ∗ 1JpA (d1J + α1)p∗Ap1J + e1α1n ∗ 1r1(K − n∗1)n∗1 (d1J + α1)p∗AKn1 − 3e1α1a1µ1n ∗ 1 d1J + α1 ] − [ α2p2J pA + α2e2a2µ2n2p ∗ 2JpA (d2J + α2)p∗Ap2J + e2α2n ∗ 2r2(K − n∗2)n∗2 (d2J + α2)p∗AKn2 − 3e2α2a2µ2n ∗ 2 d2J + α2 ] . 90 Let I1 = α1p1J pA + α1e1a1µ1n1p ∗ 1JpA (d1J + α1)p∗Ap1J + e1α1n ∗ 1r1(K − n∗1)n∗1 (d1J + α1)p∗AKn1 − 3e1α1a1µ1n ∗ 1 d1J + α1 , I2 = α2p2J pA + α2e2a2µ2n2p ∗ 2JpA (d2J + α2)p∗Ap2J + e2α2n ∗ 2r2(K − n∗2)n∗2 (d2J + α2)p∗AKn2 − 3e2α2a2µ2n ∗ 2 d2J + α2 . We have V˙3 = e1α1n ∗ 1 (d1J + α1)p∗A r1(n− n∗1)2(K − n1 − n∗1) Kn∗1n1 + e2α2n ∗ 2 (d2J + α2)p∗A r2(n− n∗2)2(K − n2 − n∗2) Kn∗2n2 − I1 − I2. Using Cauchy Schwarz inequality, we obtain I1 ≥ 0 with equality if and only if p1J p∗1J = pA p∗A and I2 ≥ 0 with equality if and only if p2Jp∗2J = pA p∗A . If ni(t) > K 2 , i = {1, 2} for t ≥ t0 leads to n∗i > K2 . Then we have r1(ni − n∗i )2(K − ni − n∗i ) Kn∗ini ≤ 0 with equality if and only if ni = n ∗ i . This implies that V˙3 ≤ 0, with equality if and only if n1 = n ∗ 1, n2 = n ∗ 2, p1J p∗1J = pA p∗A , p2J p∗2J = pA p∗A . (4.20) The invariant subsets M˜ within the set M = {(n1, n2, pA, p1J , p2J) : n1 = n∗1, n2 = n∗2, p1J p∗1J = pA p∗A , p2J p∗2J = pA p∗A } is M˜ = {(n∗1, n∗2, p∗A, p∗1J , p∗2J)}. Using Lasalle’s invariance principle (1.4.7), we obtained (n∗1, n ∗ 2, p ∗ A, p ∗ 1J , p ∗ 2J) is globally asymptotically stable on (0,+∞)5. Remark 4.2.6. The conditions for globally asymptotically stability of E4(resp. E6) are satisfied the ones for locally asymptotically stability of E4(resp. E6). Analysis and Discussion In this section, we have presented a model describing an interaction between rice and BPH in an environment diving into two patches. General analysis about non-negativity and boundedness of the model was shown. By using aggregation of variables, it was able to reduce the size of the model-obtaining a five-equation 91 systems of ordinary differential equations from a six-equation one. Then it was able to investigate the behavior of the completed model through the analysis of the reduced one. We showed global asymptotic behaviors of equilibria of the reduced model. The results correspond to different outcomes of the interaction between rice and BPH: (1) rice wins globally in competition (Theorem 4.2.2); (2) rice disappears in one of the two patches (Theorem 4.2.4); and (3) the existence of rice and BPH in both patches (Theorem 4.2.5). Figure 4.11: The case: rice wins globally in competition. Parameters values are chosen as follows: r1 = 0.7; r2 = 0.2; K = 40; a1 = 0.2; a2 = 0.2; e1 = 0.05; e2 = 0.05; α1 = 0.2; α2 = 0.3; m = 0.3; m = 0.7; d1A = 0.3; d2A = 0.5; d1J = 0.2; d2J = 0.3. We are interested in the result of Theorem 4.2.2 where BPH gets extinct. Under- standing the mechanism behind this result plays an important role for the global perspectives of rice-BPH management. To illustrate our theoretical discussion, we have provided some MATLAB simulations. We used a fourth-order Runge-Kutta method for solving our differential equations system. Now, we are going to investi- gate effects of different key factors on the extinction of BPH. Effects of dispersal parameters 92 Figure 4.12: The case: rice disappears on patch 2. Parameters values are chosen as follows: r1 = 0.7; r2 = 0.2; K = 40; a1 = 0.5; a2 = 0.7; e1 = 0.6; e2 = 0.3; α1 = 0.1; α2 = 0.2; m = 0.3; m = 0.7; d1A = 0.3; d2A = 0.5; d1J = 0.2; d2J = 0.3. Firstly, we re-write condition (4.19) of theorem 1 as follows E1x + E2(1 − x) ≤ 0 where E1 = a1e1K α1 d1J+α1 − d1A and E2 = a2e2K α2d2J+α2 − d2A. E1 (reps. E2) are the functions of birth and death rates as well as the rate describing being matures of ju- veniles. Therefore, E1 (reps. E2) can be considered as an evolution function of BPH on patch 1 (reps. patch 2). There are several following situations: If E1, E2 < 0 then condition (4.19) holds. In fact, in this case BPH does not grow on both patches. Consequently, BPH globally gets extinct. If E1, E2 > 0 then condition (4.19) does not hold. In this case, BPH grows on both patches. It is therefore survival. If E1 µ1 > E2 −E1+E2 . It means that if BPH is more likely to stay on patch 1 (where it cannot grow), it eventually get extinct. If E2 µ2 > E1 −E2+E1 . Similarly, BPH will get extinct when it distributes more enough on patch 2. Effects of age-structure parameters In order to study effects of age-structure parameters,we re-write condition (4.19) in the following way: G1µ1α1 + G2µ2α2 − (d1Jd1Aµ1 + d2Jd2Aµ2) ≤ 0 where G1 = a1e1K − d1A and G2 = a2e2K − d2A. G1 (reps. G2) are the functions of birth and death rates. That is the reason why we call G1 (reps. G2) growth function of BPH in patch 1 (reps. patch 2). If G1, G2 < 0 then (∗) is satisfied. This condition means 93 Figure 4.13: The case: the existence of rice and BPH on both patches. Parameters values are chosen as follows: r1 = 0.3; r2 = 0.9; K = 40; a1 = 0.7; a2 = 0.1; e1 = 0.9; e2 = 0.5; α1 = 0.1; α2 = 0.1; m = 0.8; m = 0.2; d1A = 0.3; d2A = 0.5; d1J = 0.1; d2J = 0.3. that when BPH’s density decreases in both patches then BPH will die as a conse- quence. If G1, G2 > 0 and (d1Jd1Aµ1 + d2Jd2Aµ2) ≈ 0 then (4.19) is not satisfied. This condition means that the energy that BPH gets from rice is higher than the loosing one due to death process. BPH is therefore survival in both patches. The present model takes into account a simple system of density-independence dispersal. It would be interesting to consider density dependent dispersals in the model. This would lead to a more complicated model and will be the subject of our future work. Conclusion We have shown, in this chapter, some models for two ecological phenomena. For the thiof-octopus system at the coast of Senegal, three models corresponding to three case with increasing complexity have proposed: (1) the case with refuge, (2) the case with refuge and density-independent migration and (3) the case with refuge and density-dependent migration. For the rice-BPH system, a model is given by the equation-based approach. We studied the local stability analysis and the global properties of their reduced models to get the knowledge about the complete model of these ecological phenomena. 94 CONCLUSION Summary of contributions Competitive ecosystems have been under investigation for a long time. Many models have been built to get the knowledge and to explain about these ecologi- cal phenomena in reality. In this thesis, we have developed some continuous and discrete models for studying the effects of the environment, the local behaviors of individuals and the age structure of population on the competitive ecosystems both in theoretical and practical point of views. The concrete results are given as follows: In term of theoretical point of view, Chapter 2 dealt with the model with two opposite behaviors (aggressive and avoiding strategies) based on migration of in- dividuals in a patchy (biotic and abiotic) environment revealed that under certain conditions, aggressiveness is efficient for survival of local inferior resource exploiter and even provokes global extinction of the local superior resource exploiter. A new methodology of graph generating from individual-based models (a case study in predation dynamics) was proposed in Chapter 3. A comparison with com- mon graphs as well as the integration in term of biology point of view were reported. In term of practical point of view, some effective models for two concrete eco- logical phenomena have been built in Chapter 4. The competition rice-brown plant hopper model with stage structure of population showed some emerge results which support for decision makers for their management. The competition thiof-octopus model coupling with fishing pressure figured out the strong increase of the fishing pressure in some areas leads to the depletion of the thiof and the invasion of its competitor, the octopus. Futures works There are numerous potential research directions that we could investigate for improving the results in this thesis. Here are some on which it would be nice to investigate. In the current models, the behaviors of species were simple introduced by taking into account of only single competitive and non-competitive patches. It would be also interesting to consider several competitive patches connected by migrations. That could lead to much more complicated model but more interesting to investigate. We considered, by using the discrete model, the competition of only two predator- prey species in a homogeneous environment. 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(accepted for publication in Vietnam Journal of Mathematical Application) 4. Thuy Nguyen-Phuong, Doanh Nguyen-Ngoc, Duong Phan-Thi-Ha, On the Generating Graph of an Individual-Based Predator-Prey Model. (submitted) 107