Global periodic solutions in a plankton-fish interaction model with toxication delay
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Amit Sharma
Abstract
In this paper, a plankton-fish interaction model is proposed and analyzed with the help of delay differential equations. Firstly, the elementary dynamical properties of the system in the absence of time delay is discussed. Then, we have established the existence of local Hopf-bifurcation as the time delay crosses its threshold value. The explicit results for stability and direction of the bifurcating periodic solution are derived by using normal form theory and center manifold arguments. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using the global Hopf-bifurcation result of [38] for functional differential equations, we establish the global existence of periodic solutions. The outcomes of the system are validated through numerical simulations in the concluding section.
1 Introduction
A major concern in population ecology is to understand how a population of a given species influences the dynamics of population of other species. The dynamic relationship between predators and their prey has long been and continued to be one of the dominant themes in both ecology as well as mathematical ecology due to its universal existence and importance [1, 2]. The initial work of [1] has established some mathematical models based on underlying ecological principles to explore the complexity of the ecological system. The research carried out in the last two decades demonstrated that very complex dynamics can arise in three or more species food chain models [3, 4, 5, 6] which may include quasi-periodicity or chaos. The perception of mechanism behind such a behavior, constitutes an exercise of paramount importance.
Introduction of strictly planktivorous fish to lakes can alter plankton communities via cascading interactions in food webs, where strong fish predation on zooplankton leads to reduction of their grazing pressure on algae. Many limnological studies have focused on this dramatic impact of fish on plankton communities [7, 8, 9, 10].
It is well recognized that toxin has great impact on plankton system and can be used as a mechanism of controlling plankton bloom [11, 12, 13, 14, 15, 16].
Recently, [17] and [18] have studied a series of mathematical models representing the prey(TPP)-specialist predator(Zooplankton)-generalist predator(Fish) interaction in the context of diagnosing and controlling deterministic chaos.
It is well known that time delay in prey-predator interactions in biological systems is a reality and it can have complex impact on the dynamic of the system namely loss of stability, induced oscillations and periodic solutions [19, 20, 21, 22, 23, 24, 25].
The interaction of plankton-nutrient model with delay due to gestation and nutrient recycling has also been studied by Ruan [27] and Das [28]. Chattopadhyay et al.[29] proposed and analyzed a mathematical model of toxic phytoplankton(Noctilucca Scintillans belonging to the group Dinoflagellates of the division Dinophyta)-zooplankton (Paracalanus belonging to the group Copepoda) interaction and assumed that the liberation of toxic substances by the phytoplankton species is not an instantaneous process but is mediated by some time lag required for maturity of species. Extending the work of [29], Bandyopadhyay et al.[30] and Rehim et al.[31] have studied the global stability of the toxin producing phytoplankton-zooplankton system. Sufficient efforts have already been made to understand the interaction of phytoplankton-zooplankton system with delay in toxin liberation, but the study of plankton-fish interaction with delay in toxin liberation by the phytoplankton species is not done so far. In this paper, an open system with three interacting components consisting of phytoplankton (p), zooplankton (z) and fish(f) is considered. Here, it is assumed that the the functional form of biomass conversion by the herbivore is of Holling type-II. The toxic substance term which causes extra mortality in zooplankton is expressed by Holling type-II functional response [32]. It is also taken into account that the liberation of toxic substances by the phytoplankton species follows discrete time variation.
In ecological sense, the study of plankton-fish interaction has practical application in the form of biomanipulation in eutrophic lakes. It is based on the premise that a reduction in the population of planktivorous fish will lead to larger zooplankton, increased grazing, and clearer water ([37] and references there in). There are many papers available in the literature where author’s have studied the plankton system with different time delays [19, 25, 29, 30, 31] and plankton-fish interaction for exploring complex dynamic in these systems (see [5, 18] and reference there in) but rare work has been found in the literature where plankton-fish interaction with toxication delay has been studied. So, main motivation behind this paper is to observe the effect of toxication delay on the plankton-fish interaction in the above context and the organization of our paper is as follows: In the next Section, the model equations for the given system are developed. The stability of the co-existence equilibrium R* in the absence of delay is discussed in Section 3. After that in Section 4, the delayed plankton model is considered and taking delay as bifurcation parameter the dynamical behavior of the system around coexisting equilibrium is discussed. The direction and stability of the bifurcating solution using a technique based upon normal form theory and center manifold theorem are determined in Section 4.1. The global existence of periodic solutions [38, 39, 40] have also been discussed in Section 5. Some support to our analytical findings through numerical simulations are given in Section 6. Finally, we mention basic outcomes of our mathematical findings and their ecological significance in the concluding Section.
2 The Mathematical Model
The following set of assumptions are assumed to formulate our mathematical model:
Let p(t) be population density of the toxin producing phytoplankton (TPP), predated by individuals of specialist predator zooplankton of population density z(t) and this zooplankton population, in turn, serves as a favorite food for the generalist predator fish of population size f(t). The tri-trophic interactions are represented by the following system of ordinary differential equations,
r be the growth rate of the TPP species, α1 be the maximum ingestion rate of zooplankton population, δ1 measures the intensity of competition between the individuals of phytoplankton species,
δ2 and δ3 be the mortality rates of phytoplankton, zooplankton and fish population, respectively,
k1 be the fraction of biomass converted to zooplankton. Let α2 be the rate at which fish graze on specialist predator (zooplankton) and k2 be the corresponding biomass conversion by fish predator. a, b and d are the half saturation constants,
Here, the effect of harmful phytoplankton species on zooplankton is modeled by Holling-II type response function,
It is assumed that the fish population is harvested quadratically at constant rate of harvesting, h,
Let τ is the time delay which is incorporated with the assumption that the liberation of toxin is not instantaneous rather it is mediated by some time lag. The biological significance of this time lag lies in the fact that this time may be considered as the time required for the maturity of toxic-phytoplankton to reduce the grazing impact of zooplankton.
The initial conditions of the system (2.1) take the form p(θ) = ϕ1(θ), z(θ) = ϕ2(θ) and f(θ) = ϕ3(θ), θ∈[−τ,0], ϕi(0) ≥ 0, ϕi(θ)∈ C([−τ,0],
3 Dynamical behavior without delay
The given system (2.1) possesses the following equilibrium points:
R0 = (0, 0, 0), a trivial equilibrium always exists,
The interior equilibrium point R* = (p*, z*, f*) of (2.1) exists if and only if conditions p1 < p* <
Lemma 3.1
The interior equilibrium R* of the given system (2.1) is invariant in positive quadrant.
Proof
To prove that for all t∈[0,M), (M > 0), p(t) > 0, z(t) > 0 and f(t) under the initial conditions p(0) > 0, z(0) > 0 and f(θ) > 0. we suppose otherwise there exists a 0 < T < M such that for all t∈[0,T), p(t) > 0, z(t) > 0 and f(t) > 0 and either p(T) = 0 or z(T) = 0 or f(T) = 0.
For any t∈[−τ,T), integration of system (2.1) yields,
Since p(t), z(t) and f(t) are both continuous in [−τ,T], there exists a S > 0 such that for all t∈[τ,T)
Taking t → T, we get p(T) > 0, z(T) > 0 and f(T) > 0, a contradiction. Thus p(t) > 0, z(t) > 0 and f(t) > 0 for any t∈[0,M).
Theorem 3.1
The system (2.1) remains uniformly bounded in ∑, where ∑ = [(p(t), z(t), f(t) : 0 ≤ p(t)+z(t)+f(t) ≤
Proof
From (2.1) for all t∈[0,∞),
or it can be obtained that, lim supt → ∞p(t) ≤
Let us consider a time dependent function,
Using (2.1) in above expression, we obtain
whereη = min(1, r, −δ2+k1α1−α1-θ).
Now, applying the theorem of differential inequalities [34], we obtain,
As t → ∞, we have 0 ≤ W(t) ≤
Hence, all the solutions of the system (2.1) are bounded in ∑.
The Jocobian matrix of the system is given by,
Proposition
At R0 = (0, 0, 0), the eigenvalues of the jacobian matrix are r, δ2, -δ3, which implies that the equilibrium E0 is an unstable point.
Remark
The dynamic around R1 = (p1, z1, 0) can be studied easily and is left for the reader.
4 Dynamical behavior with delay
Using the transformation, p = x1+p*, z = x2+z* and f = x3+f*, the linearization of given system (5.2) in the neighborhood of R* gives the following system,
where
The Jocobian matrix of the system is given by,
The characteristic equation corresponding to the matrix M is,
where
Theorem 4.1
When τ = 0, Routh-hurwitz criterion provides the characteristic equation (4.2) possesses roots with negative real parts provided B1–B4 > 0, B3–B6 > 0 and (B1−B4)(B3−B6) > 0.
The corresponding characteristic equation is given by,
To obtain the effect of digestion delay on the stability of equilibrium point R*, let us consider the case when τ ≠ 0 and the characteristic equation of this case is given by,
Now, for λ = 0, Δ(R*,τ) = B3+B6 ≠ 0, it means that λ = 0 is not a root of the characteristic equation (4.3) if the interior equilibrium point R* exist. The conditions for stability of R* when ‘τ’ varies are given by the following theorem.
Theorem 4.2
If all roots of the equation (4.3) has negative real parts and,
When H(σ, τ) = 0 has no positive root, the interior equilibrium point R* is asymptotically stable for an arbitrary delay ‘τ’,
When H(σ, τ) = 0 has at least one positive root say σ0, there exists a critical delay τ0 > 0 such that the interior equilibrium point R* is asymptotically stable for τ∈(0,τ0) and system (2.1) undergoes a Hopf-bifurcation around R* at τ = τ0,
where
Proof
Let ισ0 (σ0 > 0) is a pair of purely imaginary roots of (4.3), then we have,
Separating the real and imaginary parts, we obtain
Eliminating τ from above equations, we can write,
Setting z = σ2, (4.8) can be written as,
where
Now, from the sign of B1, B2 and B3, it can be obtained that equation (4.9) possesses a positive root say, z0 which implies (4.8) has a root σ0 =
Let us define
We now study, how the real part of the root of (4.3) vary as τ varies in a small neighborhood of τ0.
Let λ = ξ+ισ be a root of (4.3), then substituting λ = ξ+ισ in (4.5) and separating real and imaginary parts we have,
and
Thus, we can find that, G1(0,σ0,τ0) = G2(0,σ0,τ0) = 0 and |J|(0,σ0,τ0) > 0. Hence, by the implicit function theorem, G1(ξ,σ,τ) = 0 = G2(ξ,σ,τ).
Define ξ, σ as a function of τ in a neighbourhood of (0,σ0,τ0) such that ξ(τ0) = 0, σ0(τ0) = σ0,
4.1 Stability of Bifurcating Periodic Solution
In this subsection, we employ the explicit formulae suggested by [33] to determine the direction, stability and the period of the periodic solutions bifurcating from interior equilibrium R*.
Now using Taylor’s expension, (2.1) can be reduced to the following system of differential equations,
where
Let τ = τk+μ, ui(t) = ui(τ t) and ut(θ) = u(t+θ) for θ∈[−1,0] and dropping the bars for simplification of notations, system (4.11) becomes a functional differential equation in C = C([−1,0],𝔎3) as,
where u(t) = (u1(t),u2(t))T ∈ 𝔎2 and Lμ:C → 𝔎3, f:𝔎 × C → 𝔎2 are given, respectively, by
By Riesz representation theorem, there exists a function ϕ(θ,μ) of bounded variation for θ∈[−1,0] such that
In fact, we can choose
where δ denote the Dirac delta function.
For ϕ∈ C([−1,0],𝔎3), define
and
Then, system (4.12) is equivalent to
For ψ∈ C1([0,1],(𝔎3)*), define
and a bilinear inner product
where ς(θ) = ς(θ,0). Then A(0) and A* are adjoint operators. From the results of last section, we know that ±ισ0τk are the eigenvalues of A(0) and ∓ισ0τk are the eigenvalues of A*.
Let q(θ) = (1,α, β)Teισ0τkθ be the eigenvector of A(0) corresponding to the eigenvalue ιω0τk and q*(s) = D(1,α′, β′)eισ0τks be the eigenvector of A* corresponding to the eigenvalue −ισ0τk, then
It follow from the definition of A(0), Lμ(ϕ) and η(θ, μ) that [(A11+A22e−ισ0τk)−ισ0I]q(0) = 0,
where I is identity matrix of order 2, or,
On solving, we get, q(0) =
From the definition of A*, we obtain,
Since, q(θ) is the eigenvector of A* corresponding to -ισ0τk, then we have
Its solution yields,
Now, (q*, q) = d1[(1,α′,β′)(1,α,β)−
In order to have (q*, q) = 1, we can choose
such that (ψ, Aϕ) = (A*ψ, ϕ),
or −ισ0(q*, q) = (q*, Aq) = (A*q*, q) = (−ισ0q*, q) = ισ0(q*, q), i.e.(q*, q) = 0.
Next, we will compute the coordinates to describe the center manifold C0 at μ = 0. Let ut be the solution of (4.17) when μ = 0.
Define
On the center manifold C0, we have
W(t,θ) = W(z(t),z(t),θ)
where
z and z are local coordinates for center manifold C0} in the direction of q and q* respectively. Here, we consider only real solutions as W is real if ut is real. For the solution ut∈ C0 of (4.17), since μ = 0, we have
This equation can be rewritten as,
where
From (4.19) and (4.20), we have
Now, from (4.14) and (4.21), it follows that
Or
where
Comparing its coefficients with (4.21), we find
Since, W20 and W11 are in g21, we still to compute them.
Now, from (4.17) and (4.19), we have
where
Substituting (4.26) into (4.25) and comparing the coefficients, we get
From (4.25) and for θ∈[−1,0)
Using (4.21) in (4.28) and comparing coefficients with (4.26), we can obtain
and
From the definition of A(0), (4.27) and (4.29), we obtain
Solving it and for q(θ) = (1,q1,q2)Teισ0τkθ, we have
Similarly, from (4.27) and (4.30) it follows that,
where
Again from the definition of A(0) and (4.27), we have
and
where ς(θ) = ς(0,θ).
From (4.25), we know when θ = 0,
That is,
From (4.19), we have
Thus, we can obtain,
Comparing the coefficients in (4.35) and using (4.36), we get
Since ισ0τk is the eigenvalue of A(0) corresponding to eigenvector q(0), then
Substituting (4.31) and (4.37) into (4.33), we find
or
or
Similarly, substituting (4.32) and (4.38) into (4.34), we obtain,
or
Thus, we can determine W20(θ), W11(θ) from (4.31), (4.32) and g21 can be computed from (4.24).
Finally, we can compute the following quantities:
Theorem 4.1.1
μ2 determines the direction of the Hopf bifurcation: if μ2 > (μ2 < 0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for τ2 > τ20 (τ2 < τ20) ; β2 determines the stability of bifurcating periodic solutions: the bifurcating periodic solutions are orbitally asymptotically stable (unstable) if β2 < 0 (β2 > 0); and T2 determines the period of the bifurcating periodic solutions: the period increases (decreases) if T2 > 0 (T2 < 0).
5 Global continuation of periodic solutions
In earlier section, the system (2.1) is shown to undergo a Hopf-bifurcation at R* at τ = τk, i.e., a family of periodic solutions bifurcates from the positive equilibrium R*. as τ passes through the critical value τk. But the periodic solutions exist in a small neighbourhood of the critical value. In this section, we will study the global existence of periodic solutions which are obtained through local Hopf-bifurcation analysis. To study the global existence of periodic solutions, there exist two methods, one is ejective fixed point argument, the other is based on topological argument and S1-degree [38]. Here, we use a method on the global existence of periodic solutions given by Wu ([38]). For simplifications of the notations, setting z(t) = (x(t), y(t))T, we can rewrite the model (2.1) as the following functional differential equations:
where zt(θ) = (x(t + θ),y(t + θ)) ∈ C([−τ, 0],R2). Following the work of Wu, we need to define X = C([−τ, 0],R2),
Σ(F) = cl(z,τk,T)∈ X × R × R+ : z is a non-constant periodic solution of period T of the system(24)
𝔍(F) = (z, τk, T ) : F(z,τk, T ) = 0, and let
Lemma 5.1
Assume that (z, τk, T) is an isolated centre of the system (5.1) satisfying the hypothesis (A1)-(A4) in Wu [30] and Ym(z, τk, T) ≠ 0, where Ym(z, τk, T) is the mth crossing number of (z, τk, T), m is an integer ∈ J(z, τk, T). Then either (i) μ(z, τk, T) is unbounded, or (ii) μ(z, τk, T) is bounded, μ(z, τk, T)∩Σ(t) is finite and ΣYm(z, τk, T) = 0, for all m = 1, 2, … if m∈ J(z, τk, T) or it is zero otherwise.
Now, if Lemma 5.1(ii) is not true, then μ(z, τk, T) is unbounded. Hence, if it can be shown that the projections of μ(z, τk, T) onto the z-space and onto the T-space are bounded, then the projection of μ(z, τk, T) onto the τ-space will be unbounded. Further, if we can show that the projection of μ(z, τk, T) onto the τ-space is away from zero, then the projection of μ(z, τk, T) onto the τ- space must include the interval [τ, ∞). Following this ideal, we can prove our results on the global Hopf bifurcation.
Lemma 5.2
If τ is bounded, then all the periodic solutions of (4.11) are uniformly bounded.
The proof of this lemma is obvious from Theorem 3.1.
Lemma 5.3
Suppose R* exists and conditions of theorem (4.1) holds. Then the model (5.1) has no non-constant τ-periodic solution.
For a contradiction, suppose that system (5.1) has a nontrivial τ-periodic solution. Then the following system of ordinary differential equations has periodic solution
Denote,
Then,
or
Therefore, due to the Bendixson’s criterion [34], we can conclude that the system (5.2) has no periodic solution. The conclusion follows.
Theorem 5.1
Suppose all conditions of Lemma (5.3) are not holds. Let σ0 and τk, (k = 0, 1, 2, …) be defined by (4.8) and (4.9), then for each τ > τk (k ≥ 1), system (5.2) has at least k+1 periodic solutions.
Proof
The characteristic matrix of (2.1) at the equilibrium (p, z, f) is given by
where
(p, τ, T) is called a centre if F(p, τ, T) = 0 and det(Δ(z, τ, T)(t))(
Let
It is then easy to show that on (τk − ξ, τk + ξ) × ξ
Hence, the hypotheses (A1)-(A4) of Theorem 8 in Wu are satisfied. Moreover, if we define H±
then we have the crossing number of the isolated centre
Hence, it follows from Theorem of Wu that the connected component
We will now show that the projections of
6 Numerical Simulation
Here, we will give a numerical example for the validation of our results derived analytically in above sections. Consider the following system with some choice of system parameters,
Choosing initial level of population as p(t) = 3, z(t) = 1.0 and f(t) = 0.3, the given system remained LAS around R* = (15.63, 12.71, 1.52) (see fig.1 and 2) in the absence of delay (τ = 0). For τ ≠ 0, we find the values of p = 3.4729, q = -3.2571 and r = 0.0808 which indicates that the equation (4.3) has a purely imaginary root ισ0 with σ0 = 0.1612 . From (4.10), the critical value of time delay at which stability exchange takes place is given by τ0 = 4.5 such that R* remained LAS for τ∈[0, 4.5) (see fig. 3) and become unstable when τ > τ0. Numerical simulation shows that the interior equilibrium R* converges in the range 0 < τ < τ0 and loses its stability when τ passes through its critical value τ0 with the occurrence of Hopf bifurcation as depicted in fig. 3.
Convergence of system trajectories around R* at τ = 0
All trajectories converges to R* in the absence of delay i.e. at τ = 0
Hopf bifurcating periodic orbits are stable around R* at τ = 4.5
Thus, it can be seen that there is a range of parameter delay ‘τ’ such that system remains stable around interior equilibrium R* for τ < τ0 and as the magnitude of delay increases beyond τ = τ0 (keeping other parameters fixed) the system loses its stability and a limit cycle is obtained. Moreover, it also indicates that there is a threshold limit of toxication delay below which the system does not have any excitable nature and above it system shows excitability. It can be concluded that the system around the interior equilibrium R* enters into a Hopf bifurcation and exhibits the cyclic nature of blooms for a certain amount of time delay. Form practical point of view the existence of hopf bifurcation means that toxication delay can disturb the stability of plankton food chain and may last for long duration if delay increases further from its critical value.
The global continuation of periodic solution of the delayed system can be observed from the fig.4 and fig.5.
Another bifurcating stable periodic orbit around R* at τ = 20
Global continuation of periodic solutions around R* for τ = 25 > τ0
The stability determining quantities for Hopf-bifurcating periodic solutions at τ = τ0 are given by
c1(0) = −1.6307e + 009 – 6.7954e + 009i, μ2 = 1.5409e + 011, β2 = −3.2614e + 009, T2 = 3.7051e + 009.
We can conclude that the Hopf-bifurcation is supercritical in nature as well as the bifurcating periodic orbits are stable and increase as delay increases further through its critical value τ = τ0.
7 Conclusion and Discussion
The toxin released by phytoplankton species are of great interest due to its immense consequences on plankton ecology and possible detrimental effect on fisheries. It is known that planktivorous fishes can stabilizes plankton dynamics but may either promote or reduce algal biomass. In this paper, we have proposed and analyzed the dynamic of plankton-fish interaction with certain assumptions. It is assumed that the toxin liberation by the phytoplankton species follows a discrete time variation and the extra mortality caused by this toxin to zooplankton population is expressed in Holling type-II functional form. By choosing suitable values of system parameters, we have observed that given system does not show any excitable nature and remained stable in the absence of delay (τ = 0). But, there exist a certain threshold limit 0 < τ < 4.5 beyond which the stability exchange take place and system shows stable periodic oscillations with the occurrence of supercritical Hopf-bifurcation. Thus, we have observed that time delay in toxication process may cause planktonic bloom’s depicted through limit cycle oscillations. In ecological sense these results reveal that under certain parametric conditions, the toxication delay can bring instability in the planktonic food chain with the appearance and disappearance of planktonic blooms.
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Articles in the same Issue
- Frontmatter
- Magneto-thermo hydrodynamic peristaltic flow of Eyring-Powell nanofluid in asymmetric channel
- Influence of nonlinear thermal radiation on rotating flow of Casson nanofluid
- Impact of nanoparticles on flow of a special non-Newtonian third-grade fluid over a porous heated shrinking sheet with nonlinear radiation
- An algorithm based on a new DQM with modified exponential cubic B-splines for solving hyperbolic telegraph equation in (2 + 1) dimension
- Generalized Differential Transform Method for Solving RLC Electric Circuit of Non-Integer Order
- Global periodic solutions in a plankton-fish interaction model with toxication delay
- Effect of thermocapillarity and variable thermal conductivity on the heat transfer analysis of a non-Newtonian liquid thin film over a stretching surface in the presence of thermal radiation and heat source/sink
