The present study brings analytical argument on the fact that fishing pressure
can invert the outcomes of interspecific competition. As previously highlighted this
is the major ecological result considering potential fishing effects. But this result
has also practical consequences to take into account when considering ecosystem
based fishery management in general and specially in the case of emerging or new
resources in multitarget fishery context. The stock assessment and the related
scientific advice to provide for this kind of resource case cannot be based only on
classical monospecific approach. In the case of the West Africa regional emergence of
octopus high concentrations and exploitation, we have to reconsider the way to assess
the octopus resources in link with the other demersal resources and in particular the
related maximal sustanaible yields estimations on these resources. A management
based on collection of cumulated monospecifically based maximal sustanaible yields,
leads to unrealistic fisheries objectives, because based on a unrealistic vision of the
cumulated productivity of the area. This modeling reinforce the necessity to improve
our ecological knowledge and the panel tools for fisheries assessment (like models and
ecosystem indicators), in order to brings current fisheries stockbased management
towards more integrative and ecosystem based fisheries management
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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 timescale of the demography, the competition and
predation processes act on a slow timescale 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 preypredator 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 timescales 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 nonnegative 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 wellposedness of model (4.12)
In this section, we prove the wellposedness 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 nonnegativity 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 socalled slowfast 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 slowfast 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 slowfast 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 slowfast 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 slowfast 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 RouthHurwitz 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 RouthHurwitz 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 RouthHurwitz 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
nonnegativity and boundedness of the model was shown. By using aggregation
of variables, it was able to reduce the size of the modelobtaining a fiveequation
91
systems of ordinary differential equations from a sixequation 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 riceBPH management. To illustrate our theoretical discussion, we
have provided some MATLAB simulations. We used a fourthorder RungeKutta
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 rewrite 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 agestructure parameters
In order to study effects of agestructure parameters,we rewrite 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 densityindependence 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 thiofoctopus 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 densityindependent migration and (3) the case with
refuge and densitydependent migration. For the riceBPH system, a model is given
by the equationbased 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 individualbased 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 ricebrown plant
hopper model with stage structure of population showed some emerge results which
support for decision makers for their management. The competition thiofoctopus
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 noncompetitive 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. As a perspective, we would like to
consider more complex case studies of more than two species, for example, a system
95
of one prey and two predator species. We would like also to consider more complex
behaviors of individuals such as the migration behavior as well as take into account
of the dynamics of the environment in the predatorprey system.
The present competition model about rice and brown plant hopper ecological
system just took into account of the densityindependent migration. It would be
interesting to consider density dependent migrations in the model for future study.
96
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LIST OF PUBLICATIONS
1. Thuy NguyenPhuong, Doanh NguyenNgoc (2016) Effects of Behavioural Strat
egy on the Exploitative Competition Dynamics. Acta Biotheoretica, 64, pp.
495517. (SCI)
2. Thuy NguyenPhuong, Doanh NguyenNgoc, Pierre Auger, Sidy Ly, Didier
Jouffre (2016) Can Fishing Pressure Invert the Outcome of Interspecific Com
petition? The Case of the Thiof and of the Octopus Along the Senegalese
Coast. Acta Biotheoretica, 64, pp. 519536. (SCI)
3. Thuy NguyenPhuong, Oanh TranThiKim, Doanh NguyenNgoc, Effects of
Fast Dispersal and StageStructured on PredatorPrey Dynamics: A Case
Study of Brown PlantHopper Ecological System. (accepted for publication
in Vietnam Journal of Mathematical Application)
4. Thuy NguyenPhuong, Doanh NguyenNgoc, Duong PhanThiHa, On the
Generating Graph of an IndividualBased PredatorPrey Model. (submitted)
107