Chaos, Solitons and Fractals 136 (2020) 109844
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Chaos, Solitons and Fractals
Nonlinear Science, and Nonequilibrium and Complex Phenomena
journal homepage: www.elsevier.com/locate/chaos
Modelling and analysis of Limnothrissa miodon population in a Lake
Farikayi K. Mutasa a,∗, Brian Jones b, Senelani D. Musekwa-Hove a
a
b
Department of Applied Mathematics, National University of Science and Technology, P.O. Box AC939 Ascot, Bulawayo, Zimbawbwe
Department of Statistics and Operations Research, National University of Science and Technology, P.O. Box AC939 Ascot, Bulawayo, Zimbawbwe
a r t i c l e
i n f o
Article history:
Received 30 January 2020
Revised 21 April 2020
Accepted 22 April 2020
2010 MSC:
92B05
92D40
93A30
93D05
Keywords:
Limnothrissa miodon
Mathematical model
Nutrients
Phytoplankton
Stability
Zooplankton
a b s t r a c t
A mathematical model of nutrients, phytoplankton, zooplankton and Limnothrissa miodon is formulated,
analysed and simulated. The model is analyzed to gain insight into the qualitative features of the equilibrium states, which enable us to determine their stability. We employ analytical and numerical techniques to investigate the impact of nutrients and intra-specific competition on the population density of
Limnothrissa miodon. Stability analysis results agree with the simulations in that the coexistence equilibrium is locally asymptotically stable provided certain conditions are met. The coexistence equilibrium is
globally-asymptotically stable if a certain condition is met. The inflow rate of the nutrients has a positive
effect on the coexistence equilibrium, whereas the Limnothrissa miodon intra-specific competition, has a
negative effect on the coexistence equilibrium. Theoretical and numerical simulations show that the nutrients inflow rate is key to the productivity of the water body and that the population of Limnothrissa
miodon will continue to thrive in the water body as long as the nutrient inflow rate, is greater than some
threshold value.
1. Introduction
Lake Kariba (277 km long; about 5364 km2 in surface area,
160 km3 capacity; 29 m mean depth and 120 m maximum depth)
is located on the Zambezi river between latitudes 16 28 to 18 04 S
and longitudes 26 42 to 29 03 E [28]. Lake Kariba has an average
width of 19.4 km although the widest portion is 40 km [6]. The
lake is 486 m above sea level and the shoreline is approximately
2164 km [6,52]. The lake was dammed and filled in 1963 and it
was the largest man-made reservoir in the world at the time of
construction, and is today the second largest reservoir by volume
in Africa [28]. The lake is almost equally shared by the two riparian countries, Zambia and Zimbabwe and its catchment area covers 663,817 km2 extending over parts of Angola, Zambia, Namibia,
Botswana and Zimbabwe [26,28]. Nutrients enter the lake as a result of inflowing rivers and local flooding [38]. The lake experiences large outflows of between 50 and 60 km3 as compared to
its volume of 160 km3 , meaning that the lake loses large amounts
of nutrients per year [38]. Coche [7] for the study period 1965–69
obtained an average of 7.7 and 1.1 μgl−1 , from 1986 to 89 Magadza
[31–33] obtained 12.8 and 3.5 μgl−1 and Ndebele-Murisa [48] for
∗
Corresponding author.
E-mail address: farikayi.mutasa@nust.ac.zw (F.K. Mutasa).
https://doi.org/10.1016/j.chaos.2020.109844
0960-0779/© 2020 Elsevier Ltd. All rights reserved.
© 2020 Elsevier Ltd. All rights reserved.
the period 2007–09 obtained 4.2 and 4.1 μgl−1 for total nitrogen
and orthophosphate respectively in the lake. Phytoplankton feed
on nutrients and chlorophyll a is used as a proxy in the estimation of phytoplankton biomass in the lake. Phytoplankton biomass
was estimated to be in the range of 20 0–130 0 μgl−1 by Ramberg
[55] and Cronberg [8] in 1983, between 2 and 11 μgl−1 by Lindmark [25] in 1990 and 0.1–77.7 μgl−1 by Ndebele-Murisa [48] for
2007–09. Zooplankton in the lake mainly feed on phytoplankton
[27]. Zooplankton biomass in the lake was estimated to be in the
range 0.26–15.9 μgl−1 with an average of 2.76 μgl−1 by Magadza
[30], 0.0–26.9 μgl−1 by Masundire [42] for 1985–87 and 0.001–
0.522 ind./l by Ndebele-Murisa [48] for the period 2007–09. Masundire [43] estimated the zooplankton mortality to be 0.1 day−1 .
The pelagic sardine, Limnothrissa miodon (Boulenger, 1906) which
was introduced into Lake Kariba from Lake Tanganyika in 1967/8
[3], is a major food resource and a source of protein to many
people in Zimbabwe and Zambia and mostly feeds on zooplankton [27,36,37]. The natural mortality of Limnothrissa miodon is estimated to be between 0.009 and 0.0367 day−1 [1,27,44]. The Working assessment group on the assessment of Limnothrissa miodon
on Lake Kariba [1] suggested that food constraints could be also
having an effect on their natural mortality. Limnothrissa miodon
biomass has been estimated to be 90.91 kg ha−1 , 48.12 kg ha−1
and 38.66 kg ha−1 in 1981, 1982, 1983 respectively [40], 37 kg ha−1
2
F.K. Mutasa, B. Jones and S.D. Musekwa-Hove / Chaos, Solitons and Fractals 136 (2020) 109844
and 42/55 kg ha−1 in 1988 and 1992 respectively [23,24] and
16277 ± 9730 t in 2014 [29]. The biomass of zooplankton is far less
than that of phytoplankton and Limnothrissa miodon but is able to
support the Limnothrissa miodon population because it has a higher
production to biomass ration [41]. Many models have been formulated and analysed that have nutrients, phytoplankton and zooplankton [19,22,57,61].
Most nutrient, plankton and fish models differ in the functional
responses. Holling [15] classified food-limited functional responses
by type, namely Type I, II and III. In this study we use a Type I
functional form for nutrient uptake due to insufficient data needed
to estimate some parameters in either the Type II or III responses.
Gentleman and Neuheimer [14] argued that it is not easy to select
the correct functional response due to insufficient data because of
challenges of performing the necessary experiments for obtaining
the functional responses. Gentleman and Neuheimer [14] in their
simulations of the model by Franks et al. [13] showed that model
differences are not caused by satiation and that a non-satiating response does not necessarily mean stability of the dynamical system. According to Mullin [46], uncertainty in the data can allow
different functional responses to be fitted to the data and therefore there is no statistical basis for selecting one type of functional
response over another.
Pal and Chatterjee [51] in their study of a plankton-fish system used a Holling Type II to describe the feeding of zooplankton and fish on phytoplankton and zooplankton respectively. Pal
and Chatterjee [51] found out that the interior point is either stable or unstable, depending on the model parameters. The phytoplankton intrinsic growth rate and the fish mortality rate play an
important role in the change in steady states and periodic behaviour of the model [51]. A hopf bifurcation was observed by
Pal and Chatterjee [51] for a particular value of the phytoplankton carrying capacity and also for a model with time delay. Raw
et al. [56] in their plankton-fish fish model, with a predator Holing Type II response on the prey, showed the presence of a hopf
bifurcation for some value of the prey growth rate. Zhang and
Zhang [62] used time delay as a bifurcating parameter and showed
the presence of a hopf bifurcation in a plankton-fish model with
time delay. Edwards and Brindley [11] showed the existence of a
steady-state and a stable limit cycle as the bifurcating zooplankton predation parameter was varied in their nutrient-plankton
model. The nutrient, phytoplankton and zooplankton trajectories
settled to their equilibrium values after a very short time and
settled to oscillatory behaviour as the zooplankton higher predation was varied from 1 to 1.5, indicating the existence of a hopf
bifurcation.
A deterministic model that involves nutrients, phytoplankton,
zooplankton and Limnothrissa miodon has not been formulated and
analysed. In this research we intend to formulate and analyse a autonomous deterministic continuous dynamical system which consists of ordinary differential equations that describe the dynamics
of Limnothrissa miodon in the presence of nutrients, phytoplankton
and zooplankton. The basic Limnothrissa miodon model will help
in our understanding of the dynamics of the aquatic ecosystem
in Lake Kariba. Nutrients and plankton play an important role in
the dynamics of Limnothrissa miodon also referred to as kapenta
[3]. It is therefore important to investigate mathematically the
role played by nutrients, plankton and intra-specific competition
in the dynamics of Limnothrissa miodon. The studies done so far on
kapenta in Lake Kariba have mainly focused on bioeconomic analysis of the kapenta fisheries [18], correlation analysis [16,38,49],
regression analysis [5,35,49], time series analysis [9,49], surplus
production models [28,45,59] and analytical models [59]. Dynamical systems have not been used to understand how nutrients and
intra-specific competition describe and influence the dynamics of
kapenta fish populations in Lake Kariba. By formulating a math-
Fig. 1. Flow diagram of the Limnothrissa miodon model.
ematical model and analysing it, we will be able to qualitatively
explain the impact of nutrients and intra-specific competition on
the levels of kapenta fish.
This paper begins with model formulation, positivity and existence of model solutions in Section 2. The equilibrium states and
their conditions for existence are described in Section 3. Stability analysis of the steady states is done in Section 4. Numerical simulations and qualitative analysis of the model are done in
Section 5 to illustrate the dynamics of the Limnothrissa miodon
model. A detailed discussion concludes the paper in Section 6.
2. Model formulation
The model has 4 classes: N denoting the concentration of nutrients, P is the population density of phytoplankton, Z be the
zooplankton population density and L be the density of the Limnothrissa miodon population. The densities in each class are functions of time and are denoted by N(t), P(t), Z(t) and L(t) respectively. It is assumed that, nutrients enter the water body at the
rate a where a > 0 is a constant and the nutrients are depleted
naturally at a constant rate μ0 . The nutrients are depleted by phytoplankton at a rate of σ 1 NP. The growth rate of phytoplankton
is φ 1 σ 1 NP. It is assumed that the depletion rate of phytoplankton caused by mortality is proportional to P. Phytoplankton is depleted by zooplankton at a rate σ 2 PZ. The depletion of phytoplankton per unit time by zooplankton is given by σ 2 PZ is of the modified Holling’s type-I response [15], which refers to the change in
density of the phytoplankton per unit time per zooplankton as the
phytoplankton population density changes. φ 2 is the conversion
coefficient from phytoplankton into zooplankton. It is assumed that
the depletion rate of zooplankton caused by mortality is proportional to Z. The functional response of zooplankton to the Limnothrissa miodon given by σ 3 ZL is of the modified Holling’s type-I
response, which refers to the change in density of the zooplankton
per unit time per Limnothrissa miodon as the zooplankton population density changes. φ 3 is the conversion coefficient from zooplankton into Limnothrissa miodon. It is assumed that the depletion
rate of Limnothrissa miodon caused by mortality is proportional to
L and its rate of depletion caused by crowding is proportional to
L2 . The model is shown in Fig. 1.
F.K. Mutasa, B. Jones and S.D. Musekwa-Hove / Chaos, Solitons and Fractals 136 (2020) 109844
2.1. Model equations
Direct integration of (10) results in
We assume that the mixed layer in the Lake is thoroughly
mixed ∀t, so that there are no spatial gradients of concentrations
and the model system is the set of differential equations,
dN
= a − μ0 N − σ1 NP,
dt
dP
= φ1 σ1 NP − μ1 P − σ2 P Z,
dt
dZ
= φ2 σ2 P Z − μ2 Z − σ3 ZL,
dt
dL
= φ3 σ3 ZL − μ3 L − σ30 L2 .
dt
(1)
(3)
2.2. Positivity of solutions
Model system (1) describes the dynamics of an ecosystem and
it is necessary to prove that the concentrations of nutrients and the
densities of phytoplankton, zooplankton and Limnothrissa miodon
are positive for all time. For positive initial data for the ecosystem
model (1) we prove that the solutions will remain positive ∀t ≥ 0.
Theorem 1. Let the initial data be N(t) ≥ 0, P(t) ≥ 0, Z(t) ≥ 0,
L(t) ≥ 0. Then, solutions of N(t), P(t), Z(t), L(t) of system (1) are positive ∀t ≥ 0.
Proof. Considering the variable N(t) in [0, T], from the first equation of model (1) it follows that,
N (˙ t ) ≥ −μ0 N (t ) − σ1 N (t )P (t ), ∀t ∈ [0, T ].
(4)
Hence, we obtain
(−μ0 − σ1 P (s ))ds ≥ 0, ∀t ∈ [0, T ].
(5)
(6)
Direct integration of (6) results in
P (t ) ≥ P (0 ) exp
t
0
Z (˙t ) ≥ −μ2 Z (t ) − σ3 Z (t )L(t ), ∀t ∈ [0, T ].
(7)
(8)
Direct integration of (8) results in
0
t
(12)
+σ1 (φ1 − 1 )NP + σ2 (φ2 − 1 )P Z + σ3 (φ3 − 1 )ZL,
≤ a − μ0 N − μ1 P − μ2 Z − μ3 L,
≤ a − mQ (t ),
where m = min{(μ0 , μ1 , μ2 , μ3 )}. Thus,
dQ (t )
+ mQ (t ) ≤ a.
dt
(13)
Eq. (13) is a first order linear differential inequality [4], with the
solution given by
0 < Q (N, P, Z, L ) ≤
a
[1 − e−mt ] + Q (N0 , P0 , Z0 , L0 )e−mt
m
(14)
as t−→ ∞, (14) becomes
0 < Q (N, P, Z, L ) ≤
a
.
m
(15)
Therefore, all solutions of the system (1) enter the feasible region,
= (N (t ), P (t ), Z (t ), L(t )) ∈ R4+ : Q ≤
a
+ ς , ∀ς > 0 . (16)
m
This completes the proof of the theorem.
The equations in model system (1) have not been nondimensionalized, for clarity on which physical or biological effect is being considered when varying a parameter thus avoiding translation
of nondimensional parameters back into dimensional parameters.
Model system (1) has the steady states
(a) The phytoplankton free equilibrium is
(−μ1 − σ2 Z (s ))ds ≥ 0, ∀t ∈ [0, T ].
From the third equation of model (1) it follows that,
Z (t ) ≥ Z (0 ) exp
dQ
= a − μ0 N − σ1 NP + φ1 σ1 NP − μ1 P − σ2 P Z + φ2 σ2 P Z
dt
−μ2 Z − σ3 ZL + φ3 σ3 ZL − μ3 L − σ30 L2 ,
3. Equilibrium states
From the second equation of model (1) it follows that,
P (˙t ) ≥ −μ1 P (t ) − σ2 P (t )Z (t ), ∀t ∈ [0, T ].
be any solution of system (1) with
= a − μ0 N − μ1 P − μ2 Z − μ3 L − σ30 L2
to be the and mathematically feasible region. The coefficient σ 30 is
a positive constant for the crowding of the phytoplankton population. σ 1 , σ 2 , σ 3 are positive constants of proportionality. The μi ’s
for i = 0, 1, 2, 3 are depletion rate coefficients.
0
Theorem 2. A solution of model system (1) is feasible.
non-negative initial conditions.
Let Q (t ) = N (t ) + P (t ) + Z (t ) + L(t ), then
(2)
= ( N, P, Z, L ) ∈ R4 |N ≥ 0, P ≥ 0, Z ≥ 0, L ≥ 0 ,
N (t ) ≥ N (0 ) exp
(11)
Therefore solutions of system (1) with initial conditions (2) remain positive ∀t ≥ 0. (N (t ), P (t ), Z (t ), L(t )) ∈ R4
t
μ 3 L ( 0 ) e − μ3 t
≥ 0, ∀t ∈ [0, T ].
μ3 + σ30 L(0 )(1 − e−μ3 t )
Proof. It is necessary to show that system (1) is dissipative, that
4
is,
all feasible solutions are uniformly bounded in ⊂ R . Let
⎧
⎨N (0 ) = ϕ1 (0 ), P (0 ) = ϕ2 (0 ),
Z ( 0 ) = ϕ3 ( 0 ), L ( 0 ) = ϕ4 ( 0 ),
⎩
ϕi ( 0 ) > 0, i = 1, 2, 3, 4,
and define,
L(t ) ≥
2.3. Existence of solutions
The initial condition for system (1) is given by,
3
(−μ2 − σ3 L(s ))ds ≥ 0, ∀t ∈ [0, T ].
(9)
Considering the variable L(t) in [0, T], from the fourth equation
of model (1) it follows that,
L(˙t ) ≥ −L(t )(μ3 + σ30 L(t )), ∀t ∈ [0, T ].
(10)
E 1 = (N1∗ , P1∗ , Z1∗ , L∗1 ) =
a
, 0, 0, 0 .
μ0
(17)
Phytoplankton, zooplankton and Limnothrissa miodon are absent
in the water body. There are not enough nutrients to support
the growth of the phytoplankton in the ecosystem of the water
body.
(b) The zooplankton free equilibrium is
E2 = (N2∗ , P2∗ , 0, 0 ).
(18)
E2 is obtained when phytoplankton is participating in the
ecosystem, zooplankton and Limnothrissa miodon are not participating in the ecosystem. The population of phytoplankton is
4
F.K. Mutasa, B. Jones and S.D. Musekwa-Hove / Chaos, Solitons and Fractals 136 (2020) 109844
not sufficient to sustain the zooplankton population. E2 is obtained by solving the equations:
a − μ0 N − σ1 NP = 0,
(19)
φ1 σ 1 N − μ 1 = 0 .
(20)
Solving for N and P in (19) and (20) gives
N2∗ =
μ1
,
φ1 σ 1
P2∗ = aφ1 σσ11−μμ1 1 μ0 ,
(21)
Z2∗ = 0, and,
L∗2 = 0.
E2 exists provided that
aφ1 σ1 > μ1 μ0 .
μ
0
μ
to the steady state value of φ σ1 in the presence of phytoplank1 1
ton.
(c) The Limnothrissa miodon free equilibrium is
E3 = (N3∗ , P3∗ , Z3∗ , 0 ).
(23)
The population of zooplankton is not sufficient to sustain the
Limnothrissa miodon population. E3 is obtained when phytoplankton and zooplankton are participating in the ecosystem
and the Limnothrissa miodon population is not participating in
the ecosystem. E3 is obtained by solving the equations:
a − μ0 N − σ1 NP = 0,
(24)
φ1 σ 1 N − μ 1 − σ 2 Z = 0 ,
(25)
φ2 σ 2 P − μ 2 = 0 ,
(26)
(30)
Inequality (30) can be rearranged to give P2∗ > P3∗ . This shows
that the phytoplankton equilibrium value in the absence of zooplankton will be greater than the phytoplankton equilibrium
value when the zooplankton is participating in the ecosystem.
(d) The coexistence equilibrium E∗ = (N ∗ , P ∗ , Z ∗ , L∗ ) is obtained by
solving the equations:
a − μ0 N − σ1 NP = 0,
(31)
φ1 σ 1 N − μ 1 − σ 2 Z = 0 ,
(32)
φ2 σ 2 P − μ 2 − σ 3 L = 0 ,
(33)
φ3 σ3 Z − μ3 − σ30 L = 0.
(34)
a
=
,
μ0 + σ1 P3∗
μ2
,
σ 2 φ2
φ1 σ1 N3∗ − μ1
Z3∗ =
,
σ2
P3∗ =
(27)
L∗3 = 0.
μ
E3 exists if φ1 σ1 N3∗ − μ1 > 0, that is when N3∗ > φ σ1 and can
1 1
be written as N3∗ > N2∗ . This means that more nutrients are required in the ecosystem to support the presence of the zooplankton which feed on the phytoplankton. Simplifying (27) we
obtain
aσ2 φ2
=
,
μ 2 σ 1 + μ 0 σ 2 φ2
μ2
=
,
σ 2 φ2
aσ1 φ1 σ2 φ2 − (μ0 μ1 σ2 φ2 + μ1 μ2 σ1 )
Z3∗ =
,
σ 2 ( μ 2 σ 1 + μ 0 σ 2 φ2 )
P3∗
σ1 σ2 σ3 σ30 (L∗ )2 + (σ1 σ2 σ3 μ3 + σ1 σ2 μ2 σ30
+ μ0 σ22 φ2 σ30 + μ1 σ1 σ32 φ3 )L∗
+μ0 μ1 φ2 σ2 φ3 σ3 + μ1 σ1 μ2 φ3 σ3 + μ0 σ22 φ2 μ3
+ σ1 σ2 μ2 μ3 − aφ1 σ1 φ2 σ2 φ3 σ3 = 0.
(35)
Eq. (36) will have a unique positive root if the expression (37) is
positive,
A21 − 4σ1 σ2 σ3 σ30 A2 − A1
L∗ =
2σ1 σ2 σ3 σ30
>0
(36)
where,
σ1 σ2 σ3 μ3 + σ1 σ2 μ2 σ30 + μ0 σ22 φ2 σ30 + μ1 σ1 σ32 φ3 ,
A2 = μ0 μ1 φ2 σ2 φ3 σ3 + μ1 σ1 μ2 φ3 σ3 + μ0 σ22 φ2 μ3
+ σ1 σ2 μ2 μ3 − aφ1 σ1 φ2 σ2 φ3 σ3 .
(37)
A1 =
A21 − 4σ1 σ2 σ3 σ30 A2 > A21 ,
σ1 σ2 σ3 σ30 A2 < 0,
(σ1 σ2 σ3 σ30 )(μ0 μ1 φ2 σ2 φ3 σ3 + μ1 σ1 μ2 φ3 σ3 + μ0 σ22 φ2 μ3
+ σ1 σ2 μ2 μ3 − aφ1 σ1 φ2 σ2 φ3 σ3 ) < 0,
σ2 μ3 (μ0 σ2 φ2 + σ1 μ2 ) < φ3 σ3 (aφ1 σ1 φ2 σ2
− (μ0 μ1 φ2 σ2 + μ1 μ2 σ1 )),
μ3
aφ1 σ1 φ2 σ2 − (μ0 μ1 φ2 σ2 + μ1 μ2 σ1 )
<
,
φ3 σ 3
σ 2 ( μ 0 σ 2 φ2 + σ 1 μ 2 )
μ3
< Z3∗ .
(38)
φ3 σ 3
Therefore L∗ exists whenever
Z3∗ >
(28)
P∗ =
Z∗ =
L∗ =
E3 exists provided that
(29)
μ3
.
φ3 σ 3
(39)
The coexistence equilibrium is
N∗ =
L∗3 = 0.
aσ1 φ1 σ2 φ2 > μ0 μ1 σ2 φ2 + μ1 μ2 σ1 .
Solving for N, P, Z and L in (31)–(34) gives
Eq. (36) can be written as
and,
N3∗
φ2 σ2 (aφ1 σ1 − μ0 μ1 )
> μ2 .
σ 1 μ1
(22)
Rearranging the inequality (22) we obtain μa > φ σ1 . This
0
1 1
means that N1∗ > N2∗ . In the absence of phytoplankton the nua
trients will reach the value μ at equilibrium, which is reduced
N3∗
Inequality (29) can be written as
μ3 σ1 σ2 σ3 − μ2 σ1 σ2 σ30 − μ0 σ22 σ30 φ2 + μ1 σ1 σ32 φ3 +
2σ12 σ32 φ1 φ3
−μ3 σ1 σ2 σ3 + μ2 σ1 σ2 σ30 − μ0 σ22 σ30 φ2 − μ1 σ1 σ32 φ3 +
2σ1 σ22 σ30 φ2
μ3 σ1 σ2 σ3 − μ2 σ1 σ2 σ30 − μ0 σ22 σ30 φ2 − μ1 σ1 σ32 φ3 +
2σ1 σ2 σ32 φ3
−μ3 σ1 σ2 σ3 − μ2 σ1 σ2 σ30 − μ0 σ22 σ30 φ2 − μ1 σ1 σ32 φ3 +
2σ1 σ2 σ3 σ30
4A3 + A24
,
4A3 + A24
4A3 + A24
,
,
4A3 + A24
,
(40)
F.K. Mutasa, B. Jones and S.D. Musekwa-Hove / Chaos, Solitons and Fractals 136 (2020) 109844
5
(47)
where,
A3 = aσ
σ σ σ30 φ1 φ2 φ3 ,
A4 = μ3 σ1 σ2 σ3 − σ2 σ30 (μ2 σ1 + μ0 σ2 φ2 ) + μ1 σ1 σ32 φ3 .
2
1
2 2
2 3
which results in the characteristic equation
(41)
Here all the variables are participating in the water body. From
equation array (40) it follows that
μ3 σ1 σ2 σ3 − μ2 σ1 σ2 σ30 − μ0 σ22 σ30 φ2 + μ1 σ1 σ32 φ3 +
2σ12 σ32 φ1 φ3
2
∗
σ1 σ3 φ1 φ3 N − μ3 σ2 σ3 + μ2 σ2 σ30 − μ1 σ32 φ3
P∗ =
,
σ22 σ30 φ2
σ2 σ30 φ2 P∗ + μ3 σ3 − μ2 σ30
Z∗ =
,
σ32 φ3
σ3 φ3 Z ∗ − μ3
L∗ =
.
σ30
N∗ =
4A3 + A24
−σ1 N
φ1 σ 1 N − μ 1 − σ 2 Z
φ2 σ 2 Z
0
0
μ0
aφ1 σ1
μ0
− μ1
0
0
0
−μ2
0
0
aφ1 σ1
μ0
(50)
⎞
⎟
⎠.
−σ3 Z
φ3 σ3 Z − μ3 − 2σ30 L
⎞
(44)
(43)
aφ1 σ1
μ0
P3∗ ,
4.3. E3 = N3∗ , P3∗ , Z3∗ , 0
σ 1 μ1
(45)
− μ1 , λ3 = −μ2 , λ4 =
JE3
−μ0 − σ1 c2
⎜ φ1 σ 1 c 2
=⎝
0
0
(46)
−σ1 c1
0
φ2 σ 2 c 3
0
aσ φ
0
0
0
−σ2 c2
0
0
Proof. Evaluating the Jacobian matrix (43) at the equilibrium point
E2 gives,
0
(51)
μ
2
equation
2 1
0 2 2
(λ ) = (−λ3 − λ2 μ0 − λ2 c2 σ1 − λc1 c2 σ12 φ1
− λc2 c3 σ22 φ2 − c2 c3 μ0 σ22 φ2
− c22 c3 σ1 σ22 φ2 )(−λ − μ3 + c3 σ3 φ3 ).
(52)
(52) can be written as
Theorem 4. The equilibrium E2 is locally stable whenever E3 does not
exist.
JE2
⎟
⎠,
−σ3 c3
φ3 σ 3 c 3 − μ 3
(λ ) = λ4 + a1 λ3 + a2 λ2 + a3 λ + a4 = 0,
aσ 1 φ 1
−
μ1
⎜
⎜ φ1 (aφ1 σ1 − μ0 μ1 )
⎜
=⎜
μ1
⎜
⎝
0
⎞
c1 = μ σ +2μ 2σ φ ,
c2 = σ φ2
and
c3 =
2 1
0 2 2
2 2
aσ1 φ1 σ2 φ2 −(μ0 μ1 σ2 φ2 +μ1 σ1 μ2 )
. (51) results in the characteristic
σ ( μ σ +μ σ φ )
where
μ
⎛
μ
Theorem 5. The equilibrium E3 is locally stable whenever conditions
in (59) are satisfied and E∗ does not exist.
⎛
Inequality (46) can be written as μa < φ σ1 meaning that
0
1 1
∗
N1 < N2∗ , for stability of E1 . The eigenvalue λ2 is positive if
aφ 1 σ 1 > μ0 μ1 which is a condition (22) for the existence of E2 .
Therefore E1 is unstable if E2 exists. N2∗ , P2∗ , 0, 0
aφ1 σ1 −μ0 μ1
< φ σ2 meaning
2 2
that
<
for stability of E2 . More phytoplankton is required in
the ecosystem inorder to feed and sustain the zooplankton.
P2∗
Proof. Evaluating the Jacobian matrix (43) at the equilibrium point
E3 gives,
aφ1 σ1 < μ0 μ1 .
4.2. E2 =
, λ3 =
φ2 σ2 (aφ1 σ1 − μ0 μ1 )
− μ2 < 0 .
σ 1 μ1
0
0
−σ2 P
φ2 σ 2 P − μ 2 − σ 3 L
φ3 σ 3 L
− μ1 − λ )(−μ2 − λ )(−μ3 − λ ) = 0.
The eigenvalues are λ1 = −μ0 , λ2 =
−μ3 . E1 is stable if
2
(49)
0
0 ⎟
⎠,
0
−μ3
, λ2 =
aφ1 σ1 > μ0 μ1 ,
which results in the characteristic equation
(−μ0 − λ )(
2
Inequality (50) can be written as
−aσ1
0
0
φ
( μ1 1 )2 −4(aφ1 σ1 −μ0 μ1 )
1
−
(48)
aσ φ
( μ1 1 )2 −4(aφ1 σ1 −μ0 μ1 )
1
E2 exists and is stable if the following conditions are satisfied
(42)
Proof. Evaluating the Jacobian matrix (43) at the equilibrium point
E1 gives,
⎜ 0
JE1 = ⎝
aσ
− μ
1
which is the condition (30) for the existence of E3 . Therefore E2 is
unstable if E3 exists. Theorem 3. The equilibrium E1 is locally stable whenever E2 does not
exist.
−μ0
aσ1 φ1
φ2 σ2 (aφ1 σ1 − μ0 μ1 )
σ 1 μ1
φ2 σ2 (aφ1 σ1 −μ0 μ1 )
− μ2 , λ4 =
σ 1 μ1
φ2 σ2 (aφ1 σ1 −μ0 μ1 )
−μ3 . The eigenvalue λ3 is positive when
> μ2 ,
σ 1 μ1
4.1. E1 = (N1∗ , 0, 0, 0 )
⎛
The eigenvalues are λ1 =
aσ φ
− μ1 1 +
1
The stability of the equilibrium states of model (1) are determined by the Jacobian matrix [54], J , of system (1)
−μ0 − σ1 P
⎜ φ1 σ 1 P
J =⎝
0
0
aσ1 φ1
− λ ) + (aφ1 σ1 − μ0 μ1 ))(
μ1
−μ2 − λ )(−μ3 − λ ) = 0.
,
4. Stability analysis
⎛
(−λ(−
−
μ1
φ1
0
0
0
0
−σ2 (aφ1 σ1 − μ0 μ1 )
σ 1 μ1
φ2 σ2 (aφ1 σ1 − μ0 μ1 )
− μ2
σ 1 μ1
0
0
⎞
⎟
⎟
0 ⎟
⎟,
⎟
0 ⎠
−μ3
(53)
where,
a 1 = μ 0 + μ 3 + σ 1 c 2 − c 3 σ 3 φ3 ,
(54)
a2 = μ0 (μ3 − c3 σ3 φ3 ) + c2 (c3 σ22 φ2 + σ1 (μ3 + c1 σ1 φ1 − c3 σ3 φ3 ))),
(55)
a3 = c2 (c1 σ12 φ1 (μ3 − c3 σ3 φ3 ) + c3 σ22 φ2 (μ0 + μ3 + c2 σ1 − c3 σ3 φ3 ))
(56)
6
F.K. Mutasa, B. Jones and S.D. Musekwa-Hove / Chaos, Solitons and Fractals 136 (2020) 109844
a4 = c2 c3 (μ0 + c2 σ1 )σ22 φ2 (μ3 − c3 σ3 φ3 ),
(57)
a1 (a2 + a3 ) = (μ0 + μ3 + σ1 c2 − c3 σ3 φ3 )(μ0 (μ3 − c3 σ3 φ3 )
a1 > 0, a1 a2 − a3 > 0, a3 (a1 a2 − a3 ) − a21 a4 > 0, a4 > 0.
+ c2 (c1 σ12 φ1 (μ3 − c3 σ3 φ3 )
+ c2 (c3 σ22 φ2 + σ1 (μ3 + c1 σ1 φ1 − c3 σ3 φ3 ))). (58)
By the Routh-Hurwitz criterion [47], it follows that all eigenvalues of the characteristic Eq. (53) have negative real parts if,
a1 > 0, a1 a2 − a3 > 0, a3 (a1 a2 − a3 ) − a21 a4 > 0, a4 > 0.
(59)
The eigenvalue λ4 = −μ3 + c3 σ3 φ3 of (52) is positive when
μ3
.
σ 3 φ3
(60)
μ
Inequality (60) can be written as Z3∗ > σ φ3 which is the con3 3
dition (39) for the existence of E∗ . Therefore E3 is unstable if E∗
exists. 4.4. E∗ =
Remark 1. If the conditions in (46) for E1 , (49) and (50) for E2 ,
(59) for E3 and (69) for E∗ are not satisfied for a given set of parameter values, then the respective steady-state will be unstable
and there is a possibility of oscillatory behaviour for model (1).
Theorem 7. The equilibrium E∗ is globally-asymptotically stable if the
conditions in (72) are satisfied for the Lyapunov function in (70).
Proof. The proof follows Lyapunov’s second method. Let N − N ∗ >
0, P − P ∗ > 0, Z − Z ∗ > 0, L − L∗ > 0. Let V(N, P, Z, L) be a positive
Lyapunov function [20] such that V (N ∗ , P ∗ , Z ∗ , L∗ ) = 0 by,
V (N, P, Z, L ) = b1 N − N ∗ − N ∗ ln
+b3 Z − Z ∗ − Z ∗ ln
( N ∗ , P ∗ , Z ∗ , L∗ )
Theorem 6. If the equilibrium E∗ exists, then it is locallyasymptotically stable if the conditions in (69) are satisfied.
JE∗
−μ0 − σ1 c5
⎜ φ1 σ 1 c 5
=⎝
0
0
−σ1 c4
φ1 σ 1 c 4 − μ 1 − σ 2 c 6
φ2 σ 2 c 6
0
0
−σ2 c5
φ2 σ 2 c 5 − μ 2 − σ 3 c 7
φ3 σ 3 c 7
where c4 = N ∗ , c5 = P ∗ , c6 = Z ∗ and c7 = L∗ . (61) simplifies to
⎛
JE∗
−μ0 − σ1 c5
⎜ φ1 σ1 c5
=⎝
0
0
−σ1 c4
0
φ2 σ2 c6
0
0
−σ2 c5
0
φ3 σ3 c7
0
0
⎞
⎟
⎠.
−σ3 c6
φ3 σ3 c6 − μ3 − 2σ30 c7
(62)
The eigenvalues of (62) are the roots of the characteristic equation
(λ ) = c6 c7 σ32 (λ2 + λμ0 + c5 σ1 (λ + c4 σ1 φ1 ))φ3
+ (λ(λ2 + λμ0 + c5 σ1 (λ + c4 σ1 φ1 ))
+ c5 c6 (λ + μ0 + c5 σ1 )σ 2 φ2 )(λ + μ3 + 2c7 σ30 − c6 σ3 φ3 ) = 0.
(63)
(λ ) = λ + a1 λ + a2 λ + a3 λ + a4 = 0,
3
2
(64)
where,
a1 = μ0 + μ3 + c5 σ1 + 2c7 σ30 − c6 σ3 φ3 ,
(65)
a2 = c6 c7 σ32 φ3 + μ0 (μ3 + 2c7 σ30 − c6 σ3 φ3 )
+ c5 (μ3 σ1 + 2c7 σ1 σ30 + c4 σ12 φ1 + c6 σ22 φ2 − c6 σ1 σ3 φ3 ),
(66)
a3 = c4 c5 σ12 φ1 (μ3 + 2c7 σ30 − c6 σ3 φ3 ) + c6 (c52 σ1 σ22 φ2 + c7 μ0 σ32 φ3
+c5 (μ0 σ22 φ2 + μ3 σ22 φ2 + 2c7 σ22 φ2 σ30 + c7 σ1 σ32 φ3 − c6 σ22 φ2 σ3 φ3 )),
(67)
a4 = c5 c6 μ0 μ3 σ22 φ2 + c52 c6 μ3 σ1 σ22 φ2 + 2c5 c6 c7 μ0 σ22 φ2 σ30
+ b4 L − L∗ − L∗ ln
L
,
L∗
P
P∗
(70)
⎞
0
0
⎟
⎠,
−σ3 c6
φ3 σ3 c6 − μ3 − 2σ30 c7
(61)
N˙
P˙
Z˙
L˙
+ b2 ( P − P ∗ ) + b3 ( Z − Z ∗ ) + b4 ( L − L∗ ) ,
N
P
Z
L
a
∗
∗
= −b1 (N − N ) μ0 + σ1 P −
−b2 (P − P )[μ1 + σ2 Z − φ1 σ1 N]
N
∗
−b3 (Z − Z )[μ2 + σ3 L − φ2 σ2 P ] − b4 (L − L∗ )[μ3 + σ30 L − φ3 σ3 Z].
V˙ = b1 (N − N ∗ )
(71)
Then V˙ < 0 if
μ0 + σ1 P > Na , μ1 + σ2 Z > φ1 σ1 N,
μ2 + σ3 L > φ2 σ2 P, μ3 + σ30 L > φ3 σ3 Z.
(72)
Thus, in the region bounded by all points (N > N∗ , P > P∗ ,
Z > Z∗ , L > L∗ ) in (72), E∗ is globally-asymptotically stable. Numerical simulations of the model system (1) are carried out
to investigate the dynamics of the Limnothrissa miodon model using parameter values given in Table 1. The parameter values in
Table 1 are obtained from published data and others are estimates.
A fourth order Runge-Kutta numerical scheme coded in Wolfram
Mathematica is used for the numerical simulations. For model system (1) the units of the variables N, P, Z and L are μgl−1 .
For the set of the default parameters a = 1.67, μ0 = 0.0096,
σ1 = 0.3, φ1 = 1, μ1 = 0.032, σ2 = 0.12, φ2 = 1, μ2 = 0.01,
σ3 = 0.47, σ30 = 0.0 0 0 0 05, φ3 = 1, μ3 = 0.0274 the coexistence equilibrium point E∗ = (N ∗ , P ∗ , Z ∗ , L∗ ) is given by N ∗ =
0.130032, P ∗ = 42.7779, Z ∗ = 0.0584138 and L∗ = 10.9007. The condition for the existence of E∗ in (39) is satisfied since Z3∗ =
μ
120.398 and φ σ3 = 0.0582979. The eigenvalues of JE∗ are λ1 =
3 3
+2c52 c6 c7 σ1 σ22 φ2 σ30 + c4 c5 c6 c7 σ12 φ1 σ32 φ3
−c5 c62 μ0 σ22 φ2 σ3 φ3 − c52 c62 σ1 σ22 φ2 σ3 φ3 .
+ b2 P − P ∗ − P ∗ ln
5. Numerical simulations
The characteristic Eq. (63) can be written as follows:
4
Z
Z∗
N
N∗
where bi s, i = 1, 2, 3, 4 are positive constants. V is a positive definite function in the set , except at E∗ where it is zero. The rate
of change of V along the solution of system (1) is given by
Proof. Evaluating the Jacobian matrix (43) at the equilibrium point
E∗ gives,
⎛
(69)
+ c3 σ22 φ2 (μ0 + μ3 + c2 σ1 − c3 σ3 φ3 ))
c3 >
By the Routh-Hurwitz criterion, it follows that all eigenvalues
of the characteristic Eq. (64) have negative real parts if,
(68)
−12.8039, λ2 = −0.03117, λ3 = −0.00398774 − 0.420106i, λ4 =
−0.00398774 + 0.420106i. λ1 and λ2 are real and negative, λ3 and
λ4 are complex and have negative real parts, therefore E∗ is a
F.K. Mutasa, B. Jones and S.D. Musekwa-Hove / Chaos, Solitons and Fractals 136 (2020) 109844
7
Fig. 2. (a) Phase portrait showing the dynamics of zooplankton and Limnothrissa miodon; (b) Phase portrait showing the dynamics of phytoplankton and Limnothrissa miodon
for model system (1) with assumed initial condition: N (0 ) = 10, P (0 ) = 7, Z (t ) = 4, L(t ) = 2 using the default parameter values.
Fig. 3. (a) Phase portrait showing the time series of nutrients; (b) Phase portrait showing the time series of phytoplankton; (c) Phase portrait showing the time series
of zooplankton; (d) Phase portrait showing the time series of Limnothrissa miodon for model system (1) with assumed initial condition: N (0 ) = 9.5, P (0 ) = 6.5, Z (t ) =
3.5, L (t ) = 1.5 (time series with colour blue); N (0 ) = 10, P (0 ) = 7, Z (t ) = 4, L(t ) = 2 (time series with colour red); N (0 ) = 10.5, P (0 ) = 7.5, Z (t ) = 4.5, L(t ) = 2.5 (time
series with colour purple); using the default parameter values. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version
of this article.)
stable spiral. The Routh-Hurwitz conditions for local stability in
(69) are satisfied for the parameters in Table 1, therefore E∗ is locally asymptotically stable. Fig. 2(a) and (b) show the phase portraits of the zooplankton and Limnothrissa miodon dynamics and
the phytoplankton and Limnothrissa miodon dynamics for model
system (1) respectively. The phase portraits show that the coexistence equilibrium is a stable spiral which agrees with the stability analysis of the eigenvalues of JE∗ in (39). The phase portraits
show that the equilibrium point E∗ is locally asymptotically stable.
The time series plots for the nutrients, phytoplankton, zooplankton
and Limnothrissa miodon are illustrated in Fig. 3 and they show
decaying oscillations and convergence to the equilibrium state
E∗ .
The dynamics of the Limnothrissa miodon model for the nutrients, phytoplankton, zooplankton and Limnothrissa miodon populations are illustrated in Fig. 3(a)–(d) respectively. Numerical simulations of the model system (1) were carried out using a set of
parameter values given in Table 1. Fig. 3(a) illustrates the trend
for the concentration of nutrients N(t) which initially drops and
then converges asymptotically attaining an equilibrium state of
0.130032. The graph in Fig. 3(b) denotes the phytoplankton population which initially rises and eventually decays sinusoidally attaining an equilibrium state of 42.7779. The graph in Fig. 3(c) denotes the zooplankton population density which initially drops and
thereafter it converges asymptotically to an equilibrium state of
0.0584138. The graph in Fig. 3(d) denotes the population density of
8
F.K. Mutasa, B. Jones and S.D. Musekwa-Hove / Chaos, Solitons and Fractals 136 (2020) 109844
Fig. 4. (a) Time series of Limnothrissa miodon for varying σ 1 for model system (1) with σ2 = 0.56; (b) Time series of Limnothrissa miodon for varying σ 2 for model system
(1); (c) Time series of Limnothrissa miodon for varying σ 3 for model system (1) with σ2 = 0.56; (d) Time series of Limnothrissa miodon for varying μ3 for model system
(1) with σ2 = 0.56 and with assumed initial condition: N (0 ) = 10, P (0 ) = 7, Z (t ) = 4, L(t ) = 2 using a = 93 and the other default parameter values.
Limnothrissa miodon which initially rises and thereafter shows decaying oscillations and eventually converges asymptotically to the
equilibrium state of 10.9007. The nutrients, phytoplankton, zooplankton and Limnothrissa miodon populations converge asymptotically to the coexistence equilibrium point E∗ provided that conditions in (69) are satisfied. Trajectories of model system (1) converged to the same steady state E∗ from a range of initial conditions for the default parameter values as shown in Fig. 3.
Increasing the nutrient inflow rate from 1.67 to 93 μgl−1
and the zooplankton predation on phytoplankton from 0.12 to
0.56 and keeping the other default parameters constant, the coexistence equilibrium point E∗ = (N ∗ , P ∗ , Z ∗ , L∗ ) is given by N ∗ =
0.24538, P ∗ = 1263.31, Z ∗ = 0.0743107 and L∗ = 1505.2. The phytoplankton density results are in the range of the findings of Ramberg [55] and Cronberg [8]. The zooplankton density from the numerical simulations of model (1) are in the range of the findings by
L∗ =
miodon. The phytoplankton uptake rate of nutrients results in the
same equilibrium value L∗ = 1505.2 for σ1 = 0.3, 0.4 and 0.5.
Remark 2. Numerical simulations for model system (1) did not
yield a hopf bifurcation for the parameter value ranges in Table 1.
The positive stable focus obtained for a given set of parameters is
ecologically important in that a sustainable population density of
Limnothrissa miodon can be obtained by controlling harvesting of
Limnothrissa miodon.
5.1. Effects of nutrients
From (42), at the coexistence equilibrium, N∗ and L∗ are
N∗ =
μ3 σ1 σ2 σ3 − μ2 σ1 σ2 σ30 − μ0 σ22 σ30 φ2 + μ1 σ1 σ32 φ3 +
2σ12 σ32 φ1 φ3
4A3 + A24
(73)
φ3 σ3 [σ1 σ32 φ1 φ3 N∗ − μ3 σ2 σ3 + μ2 σ2 σ30 − μ1 σ32 φ3 + σ2 (μ3 σ3 − μ2 σ30 )] − σ2 μ3 σ32 φ2
,
σ2 σ32 φ3 σ30
Masundire [42]. Assuming the lake is at its capacity of 160 km3 ,
1505.2 μgl−1 of Limnothrissa miodon is 240,832 tonnes, which is
similar to 240,500 tonnes for the carrying capacity of the lake obtained by Tendaupenyu [59]. An inflow rate of 93 μgl−1 likely reflects the eutrophic phase of the lake which was a result of inundation of land and vegetation [34,60], and therefore high productivity
in the lake. After the introduction of Limnothrissa miodon into the
lake, it rapidly spread throughout the lake in a short period of time
[2].
Fig. 4 (a)–(d) illustrate the effect of varying σ 1 , σ 2 , σ 3 and μ3
in model (1) on the dynamics of Limnothrissa miodon. The effect
of higher predation values on phytoplankton and zooplankton by
the zooplankton and Limnothrissa miodon is that of increasing and
decreasing the equilibrium value of Limnothrissa miodon respectively. The effect of a higher natural mortality rate of Limnothrissa
miodon is that of decreasing the equilibrium value of Limnothrissa
,
(74)
where,
A3 = aσ12 σ22 σ32 σ30 φ1 φ2 φ3 ,
A4 =
μ3 σ1 σ2 σ3 − σ2 σ30 (μ2 σ1 + μ0 σ2 φ2 ) + μ1 σ1 σ32 φ3 .
Substituting N∗ in (73) into Eq. (74) and using the parameter
values μ0 = 0.0096, σ1 = 0.3, φ1 = 1, μ1 = 0.032, σ2 = 0.56, φ2 =
1, μ2 = 0.01, σ3 = 0.47, σ30 = 0.0 0 0 0 05, φ3 = 1, μ3 = 0.0274
given in Table 1, we obtain the functional relationship between the
Limnothrissa miodon equilibrium, L∗ , and the nutrient inflow rate, a,
as
L∗ = 1.26646 ∗ 106 (
0.0 0 0 0183537 + 1.24694 ∗ 10−7 a
−0.00428417 )
and is illustrated in Fig. 5(a). Fig. 5(b) shows the effect of varying
the inflow of nutrients into the lake on the equilibrium L∗ of the
Limnothrissa miodon.
F.K. Mutasa, B. Jones and S.D. Musekwa-Hove / Chaos, Solitons and Fractals 136 (2020) 109844
9
Fig. 5. (a) Effect of nutrient inflow rate, a, on L∗ ; (b) Time series of Limnothrissa miodon for varying a for model system (1) with assumed initial condition: N (0 ) = 10, P (0 ) =
7, Z (t ) = 4, L(t ) = 2 using parameter values in Table 1.
Table 1
Model parameters and their interpretations.
Description
Symbol
Value
Mean monthly
nutrient inflow rate
Natural depletion rate
coefficient of N
P uptake rate of N
P conversion
coefficient of N
Natural depletion rate
coefficient of P
Zooplankton grazing
rate of P
Z conversion
coefficient of P grazing
Natural depletion rate
coefficient of Z
L grazing rate of Z
Crowding effect
coefficient of L
L conversion
coefficient of Z grazing
Natural mortality of L
a
1.67 μgl−1 day−1
μ0
0.0096 day−1
σ1
φ1
0.3 lμg−1 day−1
1
μ1
0.13, 0.032 − 0.08,
0.441 day−1
0.6 − 1.4, 0.56, 0.1 −
0.69 lμg−1 day−1
0.2 − 0.75, 1.5
σ2
φ2
Source
[50]
μ2
0.01, 0.0528 day
σ3
σ 30
0.47 lμg−1 day−1
0.000005 lμg−1 day−1
φ3
1
μ3
0.0096, 0.009,
0.025–0.0367 day−1
[10,12,27]
[17,27,58]
[11,51]
−1
[21,27]
[27]
[1,27,44]
Table 2
Effect of a on the co-existence equilibrium E∗ .
Fig. 6. Plot of L∗ with the crowding effect parameter, σ 30 .
a
N∗
P∗
Z∗
L∗
0.1
5
10
0.215525
0.217303
0.219088
1.51461
76.6659
152.114
0.0583168
0.0592694
0.0602258
1.78337
91.3253
181.221
the numerical simulations. From Eq. (39) L∗ is feasible if
a>
The effect of the nutrient inflow rate, a, on the co-existence
equilibrium E∗ is shown in Table 2. L∗ doubles when a is doubled
from a = 5 to a = 10.
The partial derivative of L∗ with respect to a is
∂ L∗
=
∂a
σ1 σ2 σ3 φ1 φ2 φ3
4aσ12 σ22 σ32 σ30 φ1 φ2 φ3 + (μ3 σ1 σ2 σ3 − σ2 σ30 (μ2 σ1 + μ0 σ2 φ2 ) + μ1 σ1 σ32 φ3 )2
(75)
is positive showing that as the inflow rate, a, of nutrients increases,
the equilibrium L∗ also increases. The theoretical result agrees with
,
μ 3 σ 2 ( μ 2 σ 1 + μ 0 σ 2 φ2 ) + φ 3 σ 3 ( μ 0 μ 1 σ 2 φ2 + μ 1 σ 1 μ 2 )
. (76)
σ 1 σ 2 σ 3 φ1 φ2 φ3
Substituting the parameter values μ0 = 0.0096, σ1 = 0.3, φ1 =
1, μ1 = 0.032, σ2 = 0.56, φ2 = 1, μ2 = 0.01, σ3 = 0.47, φ3 = 1,
μ3 = 0.0274 in Table 1 into Eq. (76) we obtain a > 0.00322311. For
the parameter values given in Table 1, the coexistence equilibrium
E∗ is feasible and the Limnothrissa miodon will survive whenever
a > 0.00322311. The increase in E∗ as the nutrient inflow rate increases as illustrated in Fig. 5(a) and (b) shows that the nutrients
are the key drivers of productivity in the water body. This result
agrees with the findings of Karenge and Kolding [16] who concluded that the productivity of Lake Kariba was due to the inflow
of nutrients. Marshall [39] and Paulsen [53] showed that kapenta
abundance is correlated to river inflow and mainly influenced by
the availability of food.
10
F.K. Mutasa, B. Jones and S.D. Musekwa-Hove / Chaos, Solitons and Fractals 136 (2020) 109844
5.2. Intra-specific competition
The effect of the intra-specific competition of the Limnothrissa
miodon on L∗ is
∗
L =−
6.33232 −
(69) are satisfied. For the coexistence equilibrium point E∗ , L∗ exμ
ists whenever Z3∗ > σ φ3 . From (28) and (40) it can be seen that
3 3
Z3∗ > Z ∗ , meaning that the zooplankton settles at a higher equilib-
(0.0 0428414 − 0.0 0469056σ30 )2 + 2.3193σ30 + 0.00469056σ30 + 0.00428414
σ30
and is shown in Fig. 6. L∗ decreases as σ 30 increases.
Fig. 6 illustrates that the intra-specific competition of the Limnothrissa miodon parameter, σ 30 , has a negative effect on the Limnothrissa miodon equilibrium value, L∗ . This shows that the parameter, σ 30 , also affects the dynamical model system (1).
rium value in the absence of Limnothrissa miodon which feeds on
the zooplankton. The highlights of the study are:
•
6. Discussion
In this paper we formulated and analysed the dynamics of a
mathematical model that includes nutrients, phytoplankton, zooplankton and Limnothrissa miodon. Ordinary differential equations
are used to show the qualitative behaviour of Limnothrissa miodon
in the absence of harvesting for the food chain, nutrients → phytoplankton → zooplankton → Limnothrissa miodon. It is assumed
that that the phytoplankton growth rate, phytoplankton mortality,
grazing on phytoplankton, zooplankton growth rate, zooplankton
mortality, grazing on zooplankton and Limnothrissa miodon mortality are Holling type I forms. Theoretical analysis including positivity and existence of solutions to model (1) are investigated. We
obtained the critical points and analysed their stabilities. The local and global stability conditions of the equilibrium points are established. For local stability analysis the theoretical results agreed
with the numerical simulations in that the coexistence equilibrium is locally asymptotically stable provided certain conditions
are met. The equilibrium E∗ is globally-asymptotically stable if it
feasible and certain conditions are met. Theorem 3 shows that
when nutrients are not sufficient to support the growth of phytoplankton, then phytoplankton will be extinct. This shows that there
is a minimum value for the inflow rate of nutrients, a, which is
μ μ
necessary for the growth of phytoplankton. From (46), if a < φ0 σ 1
•
•
•
•
Theoretical results and numerical simulations of the model
agreed and show that the nutrient inflow rate and the intraspecific competition for food have a positive and negative effect
on the coexistence equilibrium point of Limnothrissa miodon respectively.
Nutrients are key to the productivity of the water body and
Limnothrissa miodon will continue to thrive as long as the nutrient inflow rate is greater than some threshold value.
The positive stable focus obtained for a given set of parameters
is ecologically important in that a sustainable population density of Limnothrissa miodon can be obtained by controlling its
harvesting.
The population density ranges for phytoplankton, zooplankton
and Limnothrissa miodon from the model are in the ranges
recorded by other authors.
In the absence of harvesting and predation, Limnothrissa miodon
will attain an equilibrium that compares well to the estimated
carrying capacity for Limnothrissa miodon in Lake Kariba.
For future studies, we intend to model the dynamics of Limnothrissa miodon with seasonal inflow of nutrients. We intend also
to analyse the population density of Limnothrissa miodon by incorporating harvesting by the fishing vessels and predation, using actual lake data. We will extend the study on Limnothrissa miodon by
incorporating environmental factors.
Funding statement
1 1
then the phytoplankton will become extinct. Theorem 4 indicates
that the population of phytoplankton is not sufficient to sustain
μ μ
the zooplankton population. From (49) and (50), if a > φ0 σ 1 and
No financial support was received from any company or organization for this study.
1 1
φ2 σ2 (aφ1 σ1 −μ0 μ1 )
− μ2 < 0 then the phytoplankton will reach its
σ 1 μ1
equilibrium value but the phytoplankton is not sufficient to sustain the zooplankton which then becomes extinct. From (49) μa >
0
μ1
∗
∗
φ1 σ1 and can be written as N1 > N2 which makes ecological sense
since the level of nutrients decrease in the presence of phytoplankφ σ (aφ σ −μ μ )
ton. Also from (50), 2 2 σ1 μ1 0 1 − μ2 < 0 can be written as
1 1
∗
∗
P2 > P3 and means that the phytoplankton settles at a higher equilibrium value in the absence of the zooplankton which grazes on
the phytoplankton. The phytoplankton will continue to thrive in
the absence of zooplankton and Limnothrissa miodon as long as if
μ μ
the nutrient inflow rate, a > σ0 φ 1 . Theorem 5 shows that more
1 1
nutrients are required in the ecosystem to support the presence
of the zooplankton which feed on the phytoplankton since from
μ
(27), φ1 σ1 N3∗ − μ1 > 0, that is when N3∗ > φ σ1 and can be written
1 1
as N3∗ > N2∗ . The inequality (30) can be rearranged to give P2∗ > P3∗ .
This shows that the phytoplankton equilibrium value in the absence of zooplankton will be greater than the phytoplankton equilibrium value when the zooplankton is participating in the ecosystem. The phytoplankton and zooplankton will continue to thrive in
the absence of Limnothrissa miodon as long as if the nutrient inflow
μ μ σ φ +μ σ
rate, a > 1 ( σ0 σ2 φ2 φ 2 1 ) . The nutrients, phytoplankton, zooplank1 2 1 2
ton and Limnothrissa miodon populations converge asymptotically
to the coexistence equilibrium point E∗ provided that conditions in
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
CRediT authorship contribution statement
Farikayi K. Mutasa: Conceptualization, Methodology, Formal
analysis, Software, Writing - original draft. Brian Jones: Supervision, Writing - review & editing. Senelani D. Musekwa-Hove: Supervision, Writing - review & editing.
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