STABILITY ANALYSIS OF NUTRITION-PHYTOPLANKTON-FIRST ZOOPLANKTON-SECOND ZOOPLANKTON- FIRST FISH-SECOND FISH (NPZ 1Z 2F 1F 2) INTERACTION MODEL

  Bulletin of Mathematics

  ISSN Printed: 2087-5126; Online: 2355-8202 Vol. 08, No. 02 (2016), pp.143-157 http://jurnal.bull-math.org

  

STABILITY ANALYSIS OF

NUTRITION-PHYTOPLANKTON-FIRST

ZOOPLANKTON-SECOND ZOOPLANKTON-

  1

  2

  1

  2 FIRST FISH-SECOND FISH (NPZ Z F F )

  

INTERACTION MODEL

_Abstract.

Asrul Sani, Kasliono and Mukhsar

  The interaction among living organism is commonly found in nature, including in the marine ecosystem, such as the relationship of producers and con- sumers in the form of competition, mutualism and predation. In this study, we develop a mathematical model describing the interactions among species in marine ecosystems, involving five marine species, i.e., phytoplankton, first zooplankton, sec- ond zooplankton, first fish, second fish, and the nutritional component. The system has eighteen equilibrium points with fifteen non-negative equilibrium points. The stability criteria for each equilibrium point were derived. Most of the equilibrium points are stable and others are unstable and saddle. Numerical experiments were conducted and they showed the behavior of the system as predicted in the analysis.

  

1. INTRODUCTION

Not only in terrestrial ecosystems, but also a food chain occurs in

marine ecosystems. The food chain shows the feeding relationship among

different living things in a particular environment or habitat involving pro-

ducers, consumers, and even decomposers [8]. The simplest part of the food

Received 16-12-2016, Accepted 27-12-2016.

  2010 Mathematics Subject Classification : 34D20, 34K28 Key words and Phrases

  : Marine ecosystem, equilibrium point, local stability, numerical experi- ments. Asrul Sani, et al. – Stability Analysis of Nutrition ...

chain is in the form of interaction between preys and predators. The food

chain in an ecosystem has served as a natural balance in which a number of

species are co-existent in the environment [4].

  If a natural unbalance occurs, for instance, the number of predators

or the consumer are more than the prey or the producer, it results in a

non-equilibrium ecosystem in which the extinction will happen [10, 4]. The

interaction among a wide range of species variety in the ocean has been sus-

tained for a long time. The interaction itself could have a positive, negative,

or no impact among species. The population dynamics of the species due

to the interaction is an interesting study.

  The marine population dynamics has been extensively studied in many

literatures. For example, Stone [14] discussed the interaction of several com-

ponents in marine ecosystem, such as bacteria, phytoplankton, zooplankton,

protozoa as well as nutrition. Pratikno and Sunarsih [10] studied the in-

teraction of three species consisting of prey, first and second predators as

a food chain model. In addition, Stone [14] studied three species of the

phytoplankton-bacteria-protozoa whereas Edward [3] presented a model of

nutrient-phytoplankton-zooplankton. Edward [2] described the relationship

between the zooplankton deaths with the dynamics of phytoplankton growth

using mathematical modeling approach. It can be seen also in Gross [4] who

discussed the role of the food chain as well as Hadley and Forbest [5] viewed

the food chain of microorganisms on marine ecosystems through mathemat-

ical model. Furthermore, the dynamics of plankton can be found in [11] and

that of algae can be seen in [15].

  Based on those basic models, Hidayatulloh and Kusumawinahyu [6]

introduced a model in a marine ecosystem with five species; nutrients, bac-

teria, phytoplankton, zooplankton and protozoa. The discussion of mathe-

matical models both in terrestrial and marine ecosystems can be found in

[1, 8, 9].

  In this study, we developed a mathematical model describing the in-

teractions among species in a marine ecosystem which involves six different

elements; phytoplankton, first zooplankton, second zooplankton, first fish,

second fish and nutrients.

  

2. MODEL FORMULATION

Suppose P denotes for the population of phytoplankton, Z for first

  1

  

zooplankton, Z for second zooplankton, F for first fish, F for second fish,

  2

  1

  2

  

and N for nutrition. Then, the model will later be called as a N P Z Z F F

• Initially, nutrients N are present at the concentration N .

• - With the presence of nutrition and the absence of predation, Phyto-

  plankton (P ) will grow exponentially with the intrinsic growth rate r > 0; dP dt

  = r

  N N P.

• - The population of phytoplankton will decrease due to predation of

  zooplankton at rate e a Z
• e i F

  1 Z

  2 ).

  1

  , respectively. In addition, only fish will be assumed to return as nutrition when they die, i.e., at rate m(d f F

  2

  and d g F

  1

  1 .

  2 F

  while first fish will be consumed by second fish at rate e h F

  2

  2 Z

  2

  Second zooplankton will be consumed by first and second fish at rate e f F

  1 .

  2 Z

  1

  1 Z

  and fish at rate e d F

  1

  2 Z

  The population of first zooplankton decreases due to the predation of second zooplankton at rate e c Z

  2 P .

  1 P + e k F

  2 P and fish at rate e g F

  1 P + e b Z

  Asrul Sani, et al. – Stability Analysis of Nutrition ... model. The dynamics of the food chain in marine ecosystem is far more complex. To simplify the real condition, several following assumptions are introduced. Assumption The assumptions relating to the interaction of the six species are as follows.

• e j F

• - First and second fish are assumed to die naturally, i.e., at rate d f F

• d g F

• - The concentration of nutrients will go down as it is consumed by phy-

  toplankton at rate mr (N/N )P .

  Schema Based on the assumptions, the scheme of the food chain of N P Z1Z2F 1F 2 can be described as in Figure 1. Equation Formulation The response function is assumed to follow Holling type I. Thus, the equation system is as follows.

  Asrul Sani, et al. – Stability Analysis of Nutrition ...

  

Figure 1: The interaction of five species with nutrient in marine ecosystem.

The arrow lines show as the predation relationship. _{8 dP N “ ”}

₁

_{2 1 2 > > > dZ > = e a Z c Z Z d F Z i F Z , > < > dZ > > dt N P − e P − e P − e P − e dt P − e − e − e 2 1} = r a Z b Z g F k F P, _{2 1 2 2 1 1 1 1 1 2 2 2 1 2}

e b Z P c Z Z f F Z j F Z ,

_{dt − e − e} = + e _{dF 1}

  (1) e F ^{1 P F 1 Z}

^{1 F}

^{1 Z 2 F 2 F} _> ^{1 F 1 ,} _{> > > − e − d dF dt 2}

= g + e d + e f h f

_{2 2 1 2 2 2 1 2 > : > > = + e + e + e > dN N dt − d} e k F P i F Z j F Z h F F g F ,

_{“ “ ” ”}

_{1 2 dt −r N} = m P + d f F + d g F .

where r as the growth rate of phytoplankton, e a and e b the grassing co-

efficient of first and second zooplankton on phytoplankton, respectively, e c

for the predation coefficient of second zooplankton on first zooplankton, e d

the predation coefficient of first fish on first zooplankton, e f the predation

coefficient of first fish on second zooplankton, e g the predation coefficient

of first fish on phytoplankton, e h , e i , e j , e k the predation coefficient of sec-

ond fish on first fish, first zooplankton, second zooplankton, phytoplankton,

respectively; d f and d g for the death rate of first concentration of nutrient.

  r r r

  Let us introduce new variables as P = p , Z = z , Z = z ,

  1

  1

  2

  2 e a e a e a r r

  

F = f , F = f , N = nN , and t = t/r , Then, the system (1)

  1

  1

  2

  2 e a e a

  becomes _{8 dp 1}

₂

_{1 1 2 >} f p, _{> > dτ np − z p − αz p − βf p − ω dz 1} = _{> =} z ^{1 2 z 1 1 z} _{> dτ} ^{1 2 f 2 z 1 ,} _{> < dz 2} p − γz − δf − ω _{2 2 1 1 2 3 2 2 df dτ − θf − ω 1} = αz p + γz z z f z , _{1 1 1 1 2 4 2 1 1} (2) βf p z z f f , _{> df > > dτ − ω − εf > 2} = + δf + θf _{1 2 2 2}

₁

_{3 2 2 4 2 1 5 2 : dn > >}

_{dτ − ω}

= ω f p + ω f z + ω f z + ω f f f , _{1 5 2}

  Asrul Sani, et al. – Stability Analysis of Nutrition ...

  e g e f d f e e e e e _{b c d k i}

  

where α = , β = , γ = , δ = , θ = , ε = , ω = , ω = ,

  1

  2 e e e e e r e e _{a a a a a a a} e d _{j e g m r h} ω = , ω = , ω = and µ = .

  3

  4

  5 e e r N e _{a a a}

  In the next section, the system of (2) will be analyzed in terms of the stability or the dynamic behaviors of its solution.

3. STABILITY ANALISYS

3.1 Equilibrium Points

  

The equilibrium points E(p, z , z , f , f , n) of (2) ore obtained by solving:

  1

  2

  1

  2

  dp dz

  1 dz 2 df 1 df 2 dn

  = = ( = = = = 0, (3) dτ dτ dτ dτ dτ dτ

There are eighteen equilibrium points as the solution of (3) but only fifteen

as non-negative equilibrium, as shown in Table 1.

• ω
• γω
¹ +ω 3<
• γω
¹ +ω 3<
• γω
¹ +ω 3<
• θ
² f ^∗ ₁ &g
• δ
² f ^∗ ₁ α δθ +γδ&
• θf ^∗
¹ δ&a
• θθ
² E
• ω ^{1 θ}
• ω ^{4 n ∗}
• ω
¹ θ−ω 3 β<
• ω
¹ θ−ω 3 β<
• γω
^{4 −ω} 3 δ<
• ω
• ω ^{2 ε +ω 4 p ∗}
• βω
• γω
^{4 −ω} 3 δ<
• γω
^{4 −ω} 3 δ<
• ω
3<
• θ _{−γε+γβp ∗ −p ∗}
• p ^∗
• ω
• ω
• γω
^{4 −ω} 3 δ<
• γω
^{4 −ω} 3 δ<
• ω
• ω
^{4 −βω} 2<
• ω
^{4 −βω} 2<
• ω
^{4 −βω} 2<
• ω ^{4 n ∗}

  3

  βp ^∗ + θω

  5

  &gt; αp ^∗ ω

  4

  1

  p ^∗ θ + ω

  3

  ε, _{−δω 5 +δω 1 p ∗ −βω 2 p ∗}

  ω ^{2 θ}

  δω

  1

  p ^∗ + ω

  2

  ε + ω

  4

  p ^∗ &gt; δω

  5

  2

  p ^∗ ,

  αp ^∗ ω ² +γω ⁵ _−γω ¹ p ^∗ _−ω ³ p ^∗ ω ^{2 θ}

  ,

  αp ^∗ ω ² +γω ⁵ γω ¹

  δ, ω

  &gt; ω

  3

  ω ⁵ αω ⁵ _−ω ^{3 n ∗}

  β

  , 0,

  ω ¹ ε−ω ^{5 β} +ω ^{4 n ∗}

  αω ⁴ +ω ¹ θ−ω ³

  β

  ,

  αω ⁴

  ω ³

  &lt; β &lt;

  ω ^{1 ε}

  αω ⁴

  4

  , ^{−αε+n ∗}

  θ αω ⁴

  and n ^∗ &gt;

  αω ⁵ ω ³ E

  16

  p ^∗ ,

  αp ^∗ ω ⁴ _−ω ¹ p ^∗ θ−ω ³

  ε +ω ³ βp ^∗ +θω ⁵ ω ^{2 θ}

  , ω

  2

  θ + γω

  &gt; p ^∗ &gt;

  γε +p ^∗ δα γβ

  δα

  E

  δω ⁵ ω ²

  &gt; ε &gt; _{−ε+δn ∗}

  δω ^{1 +ω 4} _−βω ²

  , n ^∗

  ω ^{5 β}

  ω ¹

  and

  ω ⁵ ω ²

  &gt; n ^∗ &gt;

  ε δ

  18

  2

  p ^∗ , 0, 0,

  ω ⁵ _−ω ^{1 p ∗} ω ⁴

  ,

  βp ^∗ _−ε ω ⁴

  ,

  ω ⁵ β−ω ^{1 ε}

  ω ⁴ ω ⁵

  ω ¹

  &gt; p ^∗ &gt;

  ε β

  ,

  &gt; βω

  θ ω ^{2 θ}

  ε

  , and γβω

  5

  2

  αε + θω

  5

  &gt; γω

  1

  ε + δαω

  5

  3

  γω ¹ ε−δαω ⁵ +γβω ⁵ +ω ² αε−ω ^{3 ε} +θω ⁵

  4

  ω ^{2 θ}

  or all of the symbols &gt; are exchanged with &lt; E

  17 δω ⁵ _−ω ^{2 ε}

  δω ¹

  ,

  ω ¹ ε−ω ^{5 β} +ω ^{4 n ∗}

  δω ¹

  , 0,

  ω ⁵ _−ω ^{2 n ∗} δω ¹

  , δω

  1

  αω ⁴ +ω ¹ θ−ω ³

  E

  15 θω ⁵ _−ω ^{3 ε}

  _−ω n ^∗ γ ₂ α +γω ¹ +ω ³

  6 γω ⁵ _{−ω 2 α}

  , ^{−αω 5 +ω 3 n ∗} _{−ω 2 α}

  ,

  ω ^{5 +ω 2 n ∗} _{−ω 2 α}

  , − ω

  2

  α + γω

  1

  3

  &gt; 0 and 0,

  , n ^∗

  n ^∗ ω ¹

  ω ⁵ ω ²

  &gt; n ^∗ .

  αω ⁵ ω ³ E

  7

  (0, 0, z ^∗

  2

  , 0, 0, n ^∗ ) z ^∗

  2

  , n∗ &gt; 0 E

  8

  , n ^∗ n ^∗ E

  , 0, 0, 0,

  1

  ω ⁵ ω ²

  Asrul Sani, et al. – Stability Analysis of Nutrition ...

  Tabel 1. Nonnegative Equilibrium Points

Equilibrium Points Positive (if satisfied)

E

  1

  (p ^∗ , 0, 0, 0, 0, 0) p ^∗ &gt; 0 E

  3

  p ^∗ ,

  ω ⁵ _−ω ^{1 p ∗} ω ²

  , 0, 0,

  p ^∗ ω ²

  ,

  p ^∗ &lt;

  ω ¹

  ω ⁵ ω ¹ E

  4

  p ^∗ , 0,

  ω ⁵ _−ω ^{1 p ∗} ω ³

  , 0,

  αp ^∗ ω ³

  ,

  αω ⁵ ω ³

  p ^∗ &lt;

  ω ⁵ ω ¹ E

  5 ω ⁵

  (0, z ^∗

  , 0, 0, 0, n ^∗ ) z ^∗

  ε δβ

  &gt; 0 E

  γαε

  γβ

  11 ε

  β

  , 0, 0, f ^∗

  1

  , 0, βf ^∗

  1

  f ^∗

  1

  12

  1

  δf ^∗

  1

  ,

  ε−δβf ^{∗ 1} δ

  , 0, f ^∗

  1

  , 0,

  ε δ

  f ^∗

  1

  &lt;

  &gt;

  &gt; f ^∗

  1

  1

  , n∗ &gt; 0 E

  9 θf ^∗ ₁

  α

  , 0, ^{−βθf ∗ 1 +εα}

  αθ

  , f ^∗

  1

  , 0,

  εα θ

  f ^∗

  &lt;

  γ γε

  εα βθ

  E

  10 γε _{−δα+θ+γβ}

  , ^{−γαε+γβf ∗ 1 θ−θf ∗ 1}

  δα

  (−δα+θ+γβ) γε +δ ² f ^∗ _{1 θ−θf} ^∗ ₁ δα +θ ² f ^∗ ₁

  γ (−δα+θ+γβ)

  , f ^∗

  1

  , 0, δα &lt; θ + γβ and

  f ^∗ ₁ (−δα+θ+γβ)

  Stability The Jacobian matrix of (3) is defined as:

  Asrul Sani, et al. – Stability Analysis of Nutrition ... _{  11 12}

₁₃

_{14 15 16  } a a a a a a _{21 22}

₂₃

_{24 25 26    } a a a a a a _{31 32}

₃₃

_{34 35 36  } a a a a a a J = _{ } (4) _{   a} a _{51 a} ^{41 a} _{52 a} ^{42 a} _{53 a} ^{43 a 44 a} _{54 a} ^{45 a} _{55 a 56 } ⁴⁶ a ^{61 a 62 a 63 a 64 a 65 a 66} − − −

where a = n−z αz βf ω f , a = −p, a = −αp, a = −βp, a =

  11

  1

  2

  1

  1

  2

  12

  13

  14

  15

  − − −

ω p, a = p, a = z , a = p−γz δf ω f , a = −γz , a = −δdz ,

  1

  16

  21

  1

  22

  2

  1

  2

  2

  23

  1

  24

  1

  25

  2

  1

  26

  31

  2

  32

  2

  33

  1

  1

  3

  2

  − −

a = −ω z , a = 0, a = αz , a = γz , a = αp + γz θf ω f ,

  

a = −θz , a = −ω z , a = 0, a 1 = βf , a = δf , a = θf ,

  34

  2

  35

  3

  2

  36

  4

  1

  42

  1

  43

  1

  44

  1

  2

  4

  2

  45

  4

  1

  46

  51

  1

  2

  − −

a = βp + δz + θz ω f ε, a = −ω f , a = 0, a = ω f ,

  52

  2

  2

  53

  3

  2

  54

  4

  2

  55

  1

  2

  1

  3

  2

  4

  1

  −

a = ω f , a = ω f , a = ω f , a = ω p + ω z + ω z + ω f ω ,

  5

  a

  56 = 0, a 61 = −µn, a 62 = 0, a 63 = 0, a 64 = µε, a 65 = µω 5 , a 66 = −µp.

  The stability of nonnegative equilibrium points is obtained by using

the linearization approach around those points, i.e., it is a local stability.

• _{∗ ∗ ∗ ∗ ∗}

  For the equilibrium point E . Its Jacobian matrix has the eigenvalues;

  1

  1

  2

  3

  4 _∗

  5

  − λ = −µp , λ = 0, λ = p , λ = ap , λ = βp ε, λ = −ω + ω p .

  6

  5

  1 It can be seen that λ &gt; 0 for α, p &gt; 0 whereas α and α can be

  4

  5

  6 negative. Thus, E is unstable or saddle.

  1 All eigenvalues of the Jacobian matrix evaluated in E will have nega-

• _{∗ ∗ ∗}

  5

  5

  ≤ ≤ ≤ tive real parts if it satisfies ω ω n , α n and β ω n + ǫω .

  2

  5

  3

  5

  4

  1 Thus, this equilibrium point E will be asymptotically local stable if

  5 the above conditions are satisfied.

• 7 _{∗ ∗ ∗ ∗}

  The equilibrium point E is asymptotically local stable if the following

  3

  5

  ≤ ≤ ≤ conditions are satisfied; n α?z , θz ǫ and ω z ω .

  2

  2

  2 The equilibrium point E has Jacobian matrix with positive real part

• 8

  of its eigenvalues. Therefore, E is asymptotically local unstable.

  8 The equilibrium point E 1 become asymptotically local stable if the

• 1 _{∗ ∗ ∗}

  following condition is satisfied; ǫ ≤ δf β, αǫ ≤ θf β and ω ǫ+ω f β ≤

  1

  4

  1

  1

  1 ω β.

  5 For the other equilibrium points, i.e., E , E , E , E , E 0 and E 2, their

  3

  4

  6

  9

  1

  1

  

eigenvalues have in a complex form which is difficult to write in a simple

form. Therefore, it will be studied their stability behaviors numerically

when the following conditions hold; ω α &lt; γω + ω , δα &lt; θ + γβ, and

  _{1 4} Asrul Sani, et al. – Stability Analysis of Nutrition ... ₁

  δω +ω ω ε

  &gt; β &gt; for the nonnegative equilibrium points. Let the values

  ω ^{2 ω 5}

  

of the following parameters be fixed; i.e., α = 1, β = 0.5, γ = 1, δ = 0.5,

θ = 2, ε = 4, ω = 0.5, ω = 0.5, ω = 3, ω = 0.5, ω = 6 and µ = 1. Based

  1

  2

  3

  4

  5

  

on these values, the following equilibrium points are analyzed their stability

condition. _{5 ∗} ω The equilibrium point of E should hold p &lt; = 12. For the
• 3 _∗

  ω ¹

  value p = 4.6, it is obtained that all real parts of its eigenvalues are positive which means that this equilibrium becomes asymptotically _∗ local unstable. As p increases, all real parts of its eigenvalues be- come negative. Thus, it was concluded that this point E becomes _∗

  3

  12. _∗ ω ⁵ The equilibrium point E becomes nonnegative if p &lt; = 12. For

  ≤ asymptotically local stable if 4.7 ≤ p

• 4 _∗

  ω ¹

  p = 0.1, all eigenvalues have negative real part so that the equilibrium _∗ point becomes asymptotically local stable. However, as p = 1.1, it is obtained that E becomes asymptotically local unstable. Thus, the

  4 _∗

  1.

  4 αω ^{5 ω} _∗ ⁵ The equilibrium E will be nonnegative if 2 = &lt; n &lt; = 12.

  ≤

equilibrium point E is asymptotically local stable when 0 ≤ p

• 6 _∗ ω ^{3 ω}
² E becomes asymptotically local stable for 3 ≤ n &lt; 12. For instance,

  6 _∗

  when n = 2.9 some eigenvalues have positive real parts which means _∗ that E is unstable. However, when n = 3 this equilibrium becomes

  6 stable.

• 9 _∗

  The equilibrium point E means neither first zooplankton nor second

  εα fish are present. This equilibrium will be nonnegative if f &lt; = 4. 1 βθ

  Using the fixed parameters, it is obtained the stability condition 1 &lt; _{∗ ∗} f &lt; 4. For instance, when f = 0.1 some eigenvalues have positive

  1

  1 _∗

  real parts which means that E

  9 is unstable. Meanwhile, for f = 1.1

  1 all eigenvalues have negative real parts, i.e., E is stable.

  9 The equilibrium point E

• 10 means no second fish exist in the system. _{∗ ∗}
₂ γαε δα γε f &a
• θf ^{1 +δ}
_{2 ∗} ¹ This equilibrium should hold &lt; f &lt; for non- 1 δθ _{∗ ∗} γβθ +θ +γδβ

  negative solution, i.e., 1 &lt; f &lt; 4. For f = 1.1, all real parts of

  1

  1

  eigenvalues are negative, which means that E is stable whereas for _∗

  10

  f = 3.1 it results in some eigenvalues have positive real parts and

  1 others are negative. This implies that E becomes unstable.

  10 The equilibrium E shows that second zooplankton and second fish

• 12

  are both extinct. The stability behavior of E will depend on the

  12 Asrul Sani, et al. – Stability Analysis of Nutrition ... _{∗ ∗ ∗} ε value f , i.e., f &lt; = 16. Any values of f give unstable behavior

  1 1 δβ

  1 around the point E .

  12 The equilibrium E

• 17 means that the population of second zooplankton

  ε ω _{∗ ∗} ⁵ is extinct. The point is nonnegative fore &lt; n &lt; or 8 &lt; n &lt; 12. _∗ δ ω ² For n = 8.1, it results in that all eigenvalues of this equilibrium

  point have negative real parts which means that the dynamic behavior _∗ around this equilibrium point is stable. It seems that for any n in _∗ 8 &lt; n &lt; 12 it will give a stable equilibrium point E .

  17 The equilibrium point E tells that the populations of first and second

• 18

  zooplankton are extinct and it is nonnegative equilibrium point if the _{∗ ∗ ∗ ∗} ε ω ⁵ value p belong to where &lt; p &lt; (or 8 &lt; p ) . If p = 8.1, it

  β ω ¹

  is found that this equilibrium point is unstable as in the case when _{∗ ∗ ∗}

n = 11.9. It is concluded that E for any p belong to 8 &lt; p .

  18 ₅ αω αε _∗

• 15

  The equilibrium point E should hold &lt; n &lt; for nonnega-

  ω ^{3 θ}

  tive solution. It is obtained that the dynamic behavior around this equilibrium point is stable. The equilibrium point E describes that

  16

  all species are present in the system. For nonnegative solution with _∗ ≤ the fixed parameter values, it holds the condition 1.9 ≤ p _{∗ ∗}

  2.2. For p = 1.9 and p = 2.1 it is obtained that this equilibrium point is _∗ stable. However, as p = 2.2 it gives an unstable equilibrium Thus, _∗ ≤ the stability of E on the interval 1.9 ≤ p 2.2 can be both stable

  16 and unstable.

  

4. NUMERICAL EXPERIMENT

In this study, we perform a numerical simulation for several fixed

parameters and allow one or few parameter to vary. Then, we solve nu-

merically and compare the results with those in the analysis. The fol-

lowing parameters are fixed; namely, α = 1, β = 0.5, γ = 1, δ = 0.5,

θ = 2, ε = 4, ω

  1 = 0.5, ω 2 = 0.5, ω 3 = 3, ω 4 = 0.5, ω 5 = 6 and

  

µ = 1. Using these parameter values with a range of the initial con-

centration of nutrition p, it is solved to obtain the nonnegative equilib-

rium as follows. For p = 2, we get E (2, 0, 0, 0, 0, 0), for p = 4.9, then

  1 E (4.9, 7.1, 0, 0, 9.8, 12), for p = 1, then E (1, 0, 1.833, 0, 0.333, 2), for n =

  3

  4

  

12, then E (12, 0, 0, 0, 24, 12), for n = 3, then E (2, 1, 1.5, 0, 0.99, 3), for

  5

  6

  

z = 2 and n = 1, then E (0, 0, 2, 0, 0, 1), for z = 3 and n = 4, then

  Asrul Sani, et al. – Stability Analysis of Nutrition ...

  E

  8 (0, 3, 0, 0, 0, 4), for f 1 = 1, then E 9 (2, 0, 1.5, 1, 0, 2), for f 1 = 2, then

  

E (2, 2, 1, 2, 0, 4) and E (1, 7, 0, 2, 0, 8), for f = 4 then E (8, 0, 0, 4, 0, 2),

  10

  12

  1

  11

  

for n = 9 then E (2, 7, 0, 3, 1, 9), for p = 9 then E (9, 0, 0, 3, 1, 2). Fur-

  17

  18

  

thermore, several numerical experiments around the equilibrium points are

performed to see the stability behavior of these equilibrium points.

  The dynamics of the population around E is given in Figure 2(a).

  1 Figure 2(a) shows that the dynamics of the population is away from the

  

equilibrium point E which implies that E is unstable. The numerical

  1

  1

  

simulation around the equilibrium point shows that all population does not

experience extinction.

  The equilibrium point E means that the population of second zoo-

  3

  

plankton and first fish are extinct. The trajectories of the population around

E can be seen in Figure 2(b). Figure 2(b) shows that the dynamics of the

  3

  

population approaches the equilibrium point E . Thus, it shows that equi-

  3 librium point E is stable.

  3 The dynamics of the population around the equilibrium point E is

  4

  

given in Figure 2(c). This equilibrium point E means that the population

  4

  

of first zooplankton and first fish are extinct in the long run. The trajectories

of the population approach the equilibrium point E which means that it is

  4 stable.

  Figure 2(d) shows the behavior of the solution around E . The equi-

  5

  

librium point E means that the population of first zooplankton, second

  5

  

zooplankton and first fish are extinct. Numerical simulation around the

point E shows that the population of first zooplankton, second zooplank-

  5

  

ton and first fish for the long time will experience extinction, which implies

that equilibrium point E is stable.

  5 As shown in Figure 2(e), the equilibrium point E is stable. This

  6

  equilibrium point E

  6 means that the population of first fish is extinct. Per-

  

turbation around E will initially result in fluctuation all population but all

  6

  

trajectories return to E in the long run. This numerical experiment shows

  6 that equilibrium point E is stable for the determined parameters.

  6 The dynamics of the population around the equilibrium point E is

  7

  

depicted in Figure 2(f). The equilibrium point E means that four of the

  7

  

population is extinct except second zooplankton and nutrition. Numerical

solution around the equilibrium point shows that four populations except

second zooplankton and nutrition will be extinct for the long time. It shows

that equilibrium point E is stable. Figure 2(g) shows the solution behavior

  7

  

around E . The dynamics of the population around the equilibrium point

  8 E appears to be unstable. The equilibrium point E means that all four

  8

  8 Asrul Sani, et al. – Stability Analysis of Nutrition ...

populations except first zooplankton and nutrition are extinct. However,

numerical simulation around the equilibrium point shows that all population

of do not experience extinction.

  The equilibrium point E

  9 means that the population of first zooplank-

  

ton and second fish are extinct. Figure 2(h) shows the dynamics of the

population around the equilibrium point E . numerical simulation of the

  9

  

system around E will for long run return to the equilibrium point. This

  9 indicates that it is stable.

  The dynamics of the population around the equilibrium point E , as

  10

  

shown in Figure 2(i), indicates that it is stable. This equilibrium point

E means that the population of second fish is extinct. The numerical

  10

  

experiments around this point show that the trajectories of all population

return to the equilibrium point.

  As depicted in Figure 2(j) and Figure 2(l), both the equilibrium points

E and E are stable. The equilibrium point E means that the popula-

  11

  17

  11

  

tion of first and second zooplankton and second fish are extinct, meanwhile,

the equilibrium point E means that only the population of second zoo-

  17

  

plankton is extinct. Numerical experiments around these equilibrium points

show that all trajectories in both cases return to these equilibrium in the

period of 25 unit time.

  In the case of E and E , both equilibrium point are unstable as

  12

  18

  

shown in Figure 2(k) and Figure 2(m). The point E means that the

  12

  

population of second zooplankton and second fish are extinct whereas E

  18

  

means that the population of first zooplankton and second zooplankton are

extinct. Numerical experiments around these equilibrium points show that

all trajectories in both cases are away from these points the period of 25

unit time.