# Finite Time Blowup in a Realistic Food-Chain Model

Authors: Rana D. Parshad; Hamid Ait Abderrahmane; Ranjit Kumar Upadhyay; Nitu Kumari
Journal: ISRN Biomathematics
 (2013)

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## Abstract

We investigate a realistic three-species food-chain model, with generalist top predator. The model based on a modified version of the Leslie-Gower scheme incorporates mutual interference in all the three populations and generalizes several other known models in the ecological literature. We show that the model exhibits finite time blowup in certain parameter range and for large enough initial data. This result implies that finite time blowup is possible in a large class of such three-species food-chain models. We propose a modification to the model and prove that the modified model has globally existing classical solutions, as well as a global attractor. We reconstruct the attractor using nonlinear time series analysis and show that it pssesses rich dynamics, including chaos in certain parameter regime, whilst avoiding blowup in any parameter regime. We also provide estimates on its fractal dimension as well as provide numerical simulations to visualise the spatiotemporal chaos.


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## Body

## 1. Introduction

Interaction networks in natural ecosystems can be visualized as consisting of simple units known as food chains or food webs that are made up of a number of species linked by trophic interaction [1]. A food chain model essentially comprises of the predator-prey relationship between interacting species in a given ecosystem [2]. In their seminal work [3], Hastings and Powell for the first time demonstrated that the evolution of species participating in a tritrophic relationship might be chaotic [3]. This led to a great deal of research activity in analyzing the dynamical behavior of food chain models. Upadhyay and Rai [4] provided a new example of a chaotic population system in a simple three-species food chain with Holling type II functional response. This model is different from the Hastings and Powell model, in that it considers a generalist top predator, one that can switch its food source, in the absence of its favorite prey. Letellier and Aziz-Alaoui [5] and Aziz-Alaoui [6] revisited the Upadhyay and Rai model and found that the chaotic dynamics is observed via sequences of period-doubling bifurcation of limit cycles which however suddenly break down and reverse giving rise to a sequence of period-halving bifurcation leading to limit cycles. Upadhyay in [7] next proposed a three-species food-chain model, by incorporating mutual interference, in the original model [4], thus generalizing the models in [3, 4, 8]. Parshad and Upadhyay [9] considered the diffusive form of the model proposed in [7]. Under certain restrictions on the parameter space, they proved global existence of solutions as well as the existence of a finite dimensional global attractor, for the diffusive model. The original three-species model for a generalist top predator [4] and its variants are very rich dynamically and have led to a number of works in the literature [4–9]. They continue to be recently investigated, for rich dynamical behavior, such as turing instability [10]. Despite these rich dynamics, the Parshad and Upadhyay model possesses a drawback, in that there is the possibility of finite time blowup in the model. Recall in general we say that the ∥·∥X norm of a solution blows up in finite time if
(1)limt→T*<∞∥u∥X=+∞,
where X is a suitable function space where one looks for the solution u, to the PDE or ODE at hand. Blow-up phenomenon in physical and biological problems has a rich history and has been studied in the context of gas dynamics, chemotaxis, cooperative, and competitive lotka-Volterra type systems [11–20]. Understanding blowup in a model can help to determine how robust a system is, as well as providing insight on the characteristics of the corresponding physical or biological phenomena.As the Parshad and Upadhyay model generalises many of the aforementioned three-species models, our result actually implies finite time blowup in all of those models as well. This, however, has not been reported in the literature. It is well documented in the literature [5] that in the absence of the middle predator the top predator becomes extinct if cD3<w3. This essentially means that if the middle predator is absent the top predator has to switch its food source. However, it is probably not able to predate efficiently enough, and thus the population cannot be sustained, unless the rate of sexual reproduction is above a certain threshold. If this is not so, that is, cD3<w3, then the species goes extinct. Also, it grows unboundedly if the inequality reverses. We remark that the situation can be much worse than unbounded growth and that finite time blowup is actually possible.In the current paper our primary contributions are the following.(1)
We show that the diffusive model of Parshad and Upadhyay [9] can blow up in finite time, for certain parameter regime and large enough initial data, via Theorem 2. This immediately shows the blowup of the diffusive form of the model of Upadhyay and Rai [7]. The temporal model proposed in [4] is also shown to blow up in finite time, in certain parameter regime.(2)
We propose a modification to the diffusive model and show global existence of classical solutions and global attractor, for this model, via Theorems6 and 14. The modified model is well posed and does not blow up in finite time, for any data or in any parameter regime.(3)
We reconstruct the attractor of the modified model, via nonlinear time series analysis, and show that it also exhibits rich dynamics, including chaos in certain parameter regime, just like the old model, whilst avoiding finite time blowup in any parameter regime. Numerical simulations are also provided to show the blowup in the classical models, as well as spatiotemporal chaos, in the modified model.This paper is organized as follows . In Section2, we recount the diffusive three-species food-chain model [9], that generalises several other known models in the literature. Finite time blowup results are summarized in Section 3. We propose a modification to the general model in Section 4 and show global existence of classical solutions and global attractor for this model. Various estimates made here concerning the middle predator and prey species are very similar to [21]. We reconstruct the chaotic attractor of the modified model, using nonlinear time series analysis, and estimate its fractal dimension. This, as well as some numerical simulations, is presented in Section 5. Some important conclusions are discussed in Section 6.


## 2. Food Chain Model System

In this section, we recount the model from [9]. Consider a situation where a prey population x1 is predated by individuals of population x2. The population x2 in turn serves as a favourite food for individuals of population x3. This interaction is represented by the following system of a simple prey-specialist predator-generalist predator interaction with the inclusion of spatial spread:(2)∂x1∂t=d1Δx1+a1x1-b1x12-w0x2m2(x1x1+D0)m1=g1(x1,x2,x3)+d1Δx1,(3)∂x2∂t=d2Δx2-a2x2+w1x2m2(x1x1+D1)m1-w2x3m2(x2x2+D2)m2=g2(x1,x2,x3)+d2Δx2,(4)∂x3∂t=d3Δx3+cx3m3-w3f1(x2)x3m3=g3(x1,x2,x3)+d3Δx3,
where a1, a2, b1, c, w0, w1, w2, w3, D0, D1, D2, D3≥0, f1(x2)=1/(x2+D3)∈C(R+). The parameters mi for i=1,2,3 are mutual interference parameters that model the intraspecific competition among predators when hunting for prey [22–27]. The problem is posed on Ω⊂ℝ2.  Ω is bounded, and ∂Ω is assumed to be smooth. We consider the Dirichlet boundary conditions
(5)x1=x2=x3=0on∂Ω.We also assume suitable initial conditions. In this model, prey population of sizex1 serves as the only food for the specialist predator population of size x2. This predator population, in turn, serves as a favorite food for the generalist predator population of size x3. The equations for rate of change of population size for prey and specialist predator have been written following the Volterra scheme; that is, predator population dies out exponentially in the absence of its lone prey. The interaction between this predator x2 and the generalist predator x3 is modeled by the modified version of the Leslie-Gower scheme where the loss in a predator population is proportional to the reciprocal of per capita availability of its most favorite food. a1 is the intrinsic growth rate of the prey population x1, a2 is the intrinsic death rate of the predator population x2 in the absence of the only food x1, c measures the rate of self-reproduction of generalist predator x3, and w0, w1, w2, and w3 are the maximum values which per capita growth rate can attain. b1 measures the strength of intraspecific competition among the individuals of the prey species x1. D0 and D1 quantify the extent to which environment provides protection to the prey x1 and may be thought of as a refuge or a measure of the effectiveness of the prey in evading a predator’s attack. D2 is the value of x2 at which per capita removal rate of x2 becomes w2/2. D3 represents the residual loss in x3 population due to severe scarcity of its favourite food x2. For m1=m2=1 the coefficient w0/(x1+D0) of the third term on the right-hand side of (2) is obtained by considering the probable effect of the density of the prey’s population on predator’s attack rate. If this coefficient is multiplied by x1 (the prey population at any instant of time), it gives the attack rate on the prey per predator. Denote p(x1)=(w0x1)/(x1+D0), when x1→∞, p(x1)→w0, which is the maximum that it can reach. The third term (w2x2x3)/(x2+D2) on the right-hand side of (3) represents the per capita functional response of the invertebrate predator x3 and was first introduced by Holling [28] in the ecological literature. He carried out specially designed experiments to determine the nature of these functional responses. The ecological role of per capita functional response has been well described by May [29]. The interaction terms appearing in the rate equation restore to some extent the symmetry which characterizes the Lotka-Volterra model. The generalist predator x3, in (4), is sexually reproducing species. We assume that males and females are equal in number and every individual has got equal opportunity to meet an individual of opposite sex. The first term of (4) represents growth rate of the sexually reproducing species in well-mixed conditions. w3 measures the limitation on the growth of the generalist predator x3 by its dependence on per capita availability of its most favorite prey x2.The study of ecological problems render’s one to regard communities of individuals as subpopulations affecting the survival of the individuals of other populations. One aspect of the dynamics of community interactions would be mutual interference among the interacting subpopulations, which in our model system are represented by the parametersmi,i=1,2,3. It has been shown in [24–26] that mutual interference is generally a “stabilizing” process. The parameters d1, d2, and d3 are diffusion coefficients of the populations.


## 3. Finite Time Blowup

In this section, we show that (2)–(4) blow up in finite time, for certain parameter regime and large enough initial data. In particular, the equation for x3 as described via (4) blows up. To this end we employ the first eigenvalue method for blow up,  [30]. First recall Jensen’s inequality with a probability measure.Lemma 1.
Letp(x) be a probability measure on ℝ, and let u be a nonnegative function. Then the following holds whenever f is a convex function:
(6)f(∫R‍u(x)p(x))≤∫R‍f(u(x))p(x).Note, this is easily generalised to higher dimensions. We next present our blow-up result.Theorem 2.
Consider the three-species food-chain model as described via (2)–(4). For c>(w3/D3) and initial data x30(x) large enough, that is
(7)∫Ω‍x30(x)W1(x)dx>(λ1d31(c-w3/D3))1/(m3-1),
there exists a finite time T*>0, such, that for any m3>1,
(8)limt→T*∫Ω‍x3(x,t)W1(x)dx=∞.
Here,W1(x) is a function such that -ΔW1(x)=λ1W1(x) on Ω, W1(x)=0 on ∂Ω. Furthermore, W1(x)>0 and ∫Ω‍W1(x)dx=1.Proof.
Consider the equation forx3 rewritten as
(9)∂x3∂t+w3x3m3x2+D3=d3Δx3+cx3m3.
Since x2 is assumed positive, we obtain
(10)∂x3∂t+w3x3m3D3≥d3Δx3+cx3m3.
Thus,
(11)∂x3∂t≥d3Δx3+(c-w3D3)x3m3.
Now we multiply (11) by W1, consider F(t)=∫Ω‍x3(x,t)W1(x)dx, and integrate by parts to obtain
(12)F′(t)=∫Ω‍∂x3∂tW1(x)dx≥∫Ω‍(d3Δx3+(c-w3D3)x3m3)W1(x)dx=-d3λ1F(t)+(c-w3D3)∫Ω‍x3m3W1(x)dx.
Via assumptions on the eigenfunctionW1, it is seen to be a probability measure. That is, we can define a probability measure p(x)=W1 on Ω and p(x)=0onℝ2/Ω. Now we consider the convex function f(x3)=x3m3 (note that any m3>1 works as the x3 in question is always positive). Thus, the application of Jensen’s inequality yields
(13)(c-w3D3)∫Ω‍x3m3W1(x)dx≥(c-w3D3)(∫Ω‍x3(x,t)W1(x)dx)m3=(c-w3D3)(F(t))m3.
Thus, substituting the previous equation into (12) yields
(14)F′(t)≥-d3λ1F(t)+(c-w3D3)(F(t))m3.
Simple integration in time of the previous equation yields thatF(t) blows up in finite time; that is
(15)limt→T*F(t)=limt→T*∫Ω‍x3(x,t)W1(x)dx=∞,
where
(16)T**=1d3λ1log((c-(w3/D3))(1-m3)d3λ1-(F(0))1-m31-m3)
and T*≤T**.
After some algebra, we obtain that for blowup the data has to satisfy(17)F(0)=∫Ω‍x30(x)W1(x)dx>(λ1d31(c-w3/D3))1/(m3-1).
This proves the theorem.Remark 3.
Note that the blowup ofF(t) immediately implies that the L∞ norm of r blows up as follows:
(18)limt→T*F(t)≤limt→T*∥r(x,t)∥∞∫Ω‍W1(x)dx=limt→T*∥r(x,t)∥∞=∞.Remark 4.
We also show blow-up in the Upadhyay-Rai model proposed in [4], that is, the temporal case with m1=1,m2=1, andm3=2. Also, for the model in [4], the blow-up time is given by
(19)T**=1d3λ1log((c-w3/D3)F(0)(c-w3/D3)F(0)-λ1d3).Remark 5.
We note that finite time blowup is also possible in the same weightedL1(Ω) norm, for Neumann boundary conditions. In this case, the eigenfunction W1 will have to be differently defined, that is, satisfying Neumann boundary conditions itself.


## 4. Modification to the Old Model

The aim of the current section is to introduce an improvement to (2)–(4), which is well posed. The key issue in amending the old model, (2)–(4), is to correct (4). Upadhyay and Rai introduced sexual reproduction in (4), by considering cx32. Under this assumption, the number of males and females in the population is assumed to be equal, [5]. Thus, mating takes place as c×(1/2)x3×(1/2)x3, and consequently the population grows at rate c. We point out that this assumption may fail in various structured populations, where due to specific demographics, the numbers of females and males are rarely equal. We modify this by considering the mating term to be cx32-ϵ=cx3×x3(1-ϵ), where we restrict 0<ϵ<1. This modification may be interpreted as modeling a population where the number of males is different from the number of females (even if very slightly so). If the numbers of males and females are in fact equal or close to, one can choose 0<ϵ≪1. Next, the mutual interference parameter is incorporated, by raising the reaction terms to the m3 power. The primary role of the previously stated modification is that now the reaction term in (4) is cx3m3-w3f1(x2)x3m4, and because we enforce m3<m4, standard parabolic theory [31] guarantees global existence. Based on the above, we introduce the following three-species food-chain model, with a generalist top predator,
(20)∂x1∂t=d1Δx1+a1x1-b1x12-w0x2m2(x1x1+D0)m1,(21)∂x2∂t=d2Δx2-a2x2+w1x2m2(x1x1+D0)m1-w2(x2x2+D2)m2x3m2,(22)∂x3∂t=d3Δx3+cx3m3-w3f1(x2)x3m4.The problem is posed onΩ⊂ℝ2. Ω is bounded, and ∂Ω is assumed to be smooth. We consider the Dirichlet boundary conditions
(23)x1=x2=x3=0on∂Ω.Note, the proofs of global existence to follow will work the with Neumann boundary conditions as well. The forms of (2)-(3) are kept essentially the same; however, we assume that D0=D1, as commonly done in numerical simulations, [5]. We also assume 0<m2≤1. This facilitates the estimates to follow. Restricting m3 to the range 1<m3<m4 has distinct advantages over the old model (2)–(4). Firstly, finite time blowup is prevented, and this is independent of the sign of c-w3/D3. Secondly, in the absence of x2, if c-w3/D3<0, the dynamics of (2)–(4) is quite boring, and x3 decays to 0, [5, 6]. However, if 1<m3<m4, even if x2=0 and c-w3/D3<0, x3 can still persist. Thus, this formulation plausibly models a true generalist x3. One that does not go extinct, even if its favorite food source does.


### 4.1. Preliminaries

The system (20)–(22) is an example of a good semilinear parabolic system, and the standard theory for parabolic systems [31] ensures the existence of a nonnegative local solution. Next, comparison arguments can be used to derive global existence for initial data in L∞(Ω). We assume that x1, x2, and x3 are nonnegative because they are densities. This assumption ensures that x1+D0>0,  x1+D1>0,  x2+D2>0,  and x2+D3>0. The nonlinearities are thus well defined. Standard parabolic theory guarantees the uniform boundedness of x3. We state the following theorem.Theorem 6.
Consider the three-species food-chain model described by (20)–(22). For any positive initial data (x10,x20,x30)∈L2(Ω) and 1<m3<m4, 0<m2≤1, there exists a global classical solution (x1,x2,x3) to the system.Remark 7.
In this section, we demonstrate a stronger version of global existence. That is, we derive uniformL∞(Ω) bounds on the solution with the initial data only in L2(Ω).The opposite signed nonlinearities here cause problems in making directH1 estimates and proving the existence of bounded absorbing sets, for the x2 variable. To circumvent this difficulty, we resort to grouping estimates via appropriate addition of the equations at hand. This technique is very similar to what we adopted in [21]. This condition plays a key role in the dissipation process. If it is satisfied, there is often a global attractor. We will also show the existence of a global attractor for the reaction diffusion system (20)–(22).Remark 8.
Note, in [9] in order to proceed, we need to assume c≤w3/(h+v). Thus, our results in [9] are only true in the parameter regime, such that v<(w3/c)-D3. Furthermore in [9], strong L∞ boundedness requirements on the solutions are assumed. Here, we enforce no such requirement, and the methods of analysis are quite different from [9].Definition 9.
Consider a semigroup,S(t), acting on a reflexive Banach space, H. Then, the global attractor, 𝒜⊂H, for this semigroup is an object that has the following properties: (i)
𝒜 is compact in H,(ii)
𝒜 is invariant; that is, S(t)𝒜=𝒜,t≥0,(iii)
ifB is bounded in M, then distM(S(t)B,𝒜)→0, t→∞.
Next, various preliminaries are presented, detailing the phase spaces of interest and recalling certain standard theories. Let us define our phase spaces of interest as(24)H=[L2(Ω)]3,V=[H1(Ω)]3.
To prove the existence of a global attractor, we are required to show [32](i)
the existence of a bounded absorbing set in the phase space;(ii)
the asymptotic compactness property of the semigroup in question. These are defined next.Definition 10 (bounded absorbing set).
A bounded set,ℬ, in a reflexive Banach space, H, is called a bounded absorbing set if, for each bounded subset U of H, there is a time, T=T(U), such that S(t)U⊂ℬ for all t>T. The number T=T(U) is referred to as the compactification time for S(t)U. This is essentially the time after which the semigroup compactifies.Definition 11 (asymptotic compactness).
The semigroup{S(t)}t≥0:H→H associated with a dynamical system is said to be asymptotically compact in H if, for any {un(0)}n=1∞ bounded in H and a sequence of times {tn→∞},   S(tn)un(0) possesses a convergent subsequence in H.


### 4.2. Global Existence and Existence of Absorbing Sets inL2(Ω)

We begin by multiplying (20) by x1 and integrating by parts over Ω. This yields
(25)12ddt∥x1∥22=-d1∥∇x1∥22+a1∥x1∥22-b1∥x1∥33-w0∫Ω‍x1m1+1x2m2(x1+D0)m1dx.We then use Hölder’s inequality followed by Young’s inequality to obtain(26)12ddt∥x1∥22≤-d1∥∇x1∥22+b1∥x1∥33+C-b1∥x1∥33-w0∫Ω‍x1m1+1x2m2(x1+D0)m1dx.
Next, using the compact Sobolev embedding, H1(Ω)↪L2(Ω), and the positivity of x1 and x2, we find that
(27)ddt∥x1∥22+C1∥x1∥22≤C,
where C depends explicitly on b1 and a1.Thus, application of Gronwall’s Lemma gives us the following estimate:(28)∥x1∥22≤e-C1t∥x1(0)∥22+CC1.Therefore, there exists timet1 given explicitly by
(29)t1=max(0,ln(∥x1(0)∥22)C1),
such that, for all t≥t1, the following estimate holds uniformly:
(30)∥x1∥22≤1+C2C1.We now make a local in time estimate for∇x1. By integrating (26) in the time interval [t1,t1+1], we obtain
(31)∫t1t1+1‍∥∇x1∥22dt≤∥x1(t1)∥22+∫t1t1+1‍C2dt≤C.
Thus, via a mean value theorem for integrals, there exists a time t2∈[t1,t1+1] such that the following estimate holds:
(32)∥∇x1(t2)∥22≤C.We now move on to showing the existence of an absorbing set forx2 in L2(Ω). By multiplying (20) by w1 and (21) by w0, adding the two together, and setting w=w1x1+w0x2, we obtain
(33)∂w∂t=d2Δw+(d1-d2)w1Δx1+w1a1x1-w1b1x12-w0a2x2-w0w2(x2x1+D2)m2x3m2.
By multiplying (33) by w, integrating by parts over Ω, and using Hölder’s inequality followed by Young’s inequality, keeping in mind the positivity of the solutions, we find that
(34)12ddt∥w∥22+d2∥∇w∥22≤w1a1∫Ω‍x1wdx+∫Ω‍(d2-d1)w1∇x1·∇wdx≤C∥w∥22+C2∥x1∥22+C1∥∇x1∥22+d22∥∇w∥22.
Thus, by using the compact Sobolev embedding, H1(Ω)↪L2(Ω), we obtain
(35)ddt∥w∥22+C∥w∥22≤C1∥∇x1∥22+C2∥x1∥22.
Next, we multiply the previous equation by eCt and integrate in time from 0 to t to find
(36)∥w∥22≤e-Ct∥w(0)∥22+∫0t‍e-(t-τ)∥∇x1∥22dτ+C1(x10,t),
where C1(x10,t)=e-Ct∫0t‍e-C1τ∥x1(0)∥22dτ+(C/C1). This follows via (30).We now make the following estimate via time integration of (26), after multiplying through by eCt:
(37)∫0t‍eCτ∥∇x1∥22dτm≤CeCt-12∫0t‍(eτddτ∥x1∥22)m=CeCt-12[eCt∥x1(t)∥22-∥x10∥22-∫0t‍eCτ∥x1(τ)∥22dτ]m≤CeCt+∥x10∥22+∫0t‍eCτ∥x1(τ)∥22dτm≤CeCt+∥x10∥22.
This follows via (30).We now substitute the above into (36) and use the form of C1(x10,t) to obtain
(38)∥w∥22≤e-Ct∥w(0)∥22+C1(x10,t)+Ce-Ct(CeCt+∥x10∥22).
Therefore, there exists time t3, given explicitly by
(39)t3=max(0,ln(w1∥x1(0)∥22+w0∥x1(0)∥22)C),
such that, for all t≥t3, the following estimate holds uniformly:
(40)∥x2∥22≤1w0∥w∥22≤1+C+CC1.
Therefore,
(41)limsupt→∞∥x2∥22≤1w0limsupt→∞∥w∥22≤C,
where C is a constant that is independent of the time and the initial data.We next derive local in time estimates for∇x2. We proceed by multiplying (21) by x2 and integrating by parts over Ω to obtain
(42)12ddt∥x2∥22=-d2∥∇x2∥22-a2∥x2∥22+w1∫Ω‍(x1(x1+D0))m1x2m2+1dx-w2∫Ω‍x2m2+1(x2+D2)m2x3m2dx.We now use Hölder’s inequality, positivity of solutions, the embeddingL2↪Lm2+1 (via assumption on m2), and the earlier estimate via (40) to yield
(43)ddt∥x2∥22+d2∥∇x2∥22≤∥x1x1+D0∥∞m1∥x2∥22≤C1.
By integrating this inequality in the time interval [t,t+1], for t≥t3, we obtain
(44)∫tt+1‍∥∇x2∥22ds≤∥x2(t)∥22+∫t1t1+1‍C1|Ω|ds≤C.
The existence of an absorbing set for x3 in L2(Ω) follows easily via the form of (22).Standard estimates imply the existence of a timet5 given explicitly by
(45)t5=max(0,ln(|x3(0)|22)C),
such that, for all t≥t5, the following estimates hold uniformly:
(46)∥x3∥22≤1+C2C,∫tt+1‍∥∇x3∥22ds≤C.We can now state the following lemma.Lemma 12.
Consider(x1,x2,x3) which are solutions to the diffusive three-species food-chain model described via (20)–(22), with (x10,x20,x30)∈L2(Ω), 1<m3<m4, and  0<m2≤1. There exists a time  t*=max(t1,t3,t5) and a constant C independent of time and the initial data and dependent only on the parameters in (20)–(22), such that, for any t>t*, the following uniform estimates hold:
(47)∥x1∥22≤C,∥x2∥22≤C,∥x3∥22≤C.We now make theH1 estimate. We take the inner product of (20) with -Δx1 and integrate by parts over Ω. Thus, we obtain
(48)12ddt∥∇x1∥22+d1∥Δx1∥22+2b1∫Ω‍x1|∇x1|2dx=a1∥∇x1∥22+w0∫Ω‍(x1(x1+D0))m1(-Δx1)(x2)m2dx.
We apply the Cauchy-Schwartz inequality, the Cauchy with epsilon, and Young’s inequalities on the last term, along with the embedding L2↪L2m2 (via assumption on m2) to obtain
(49)12ddt∥∇x1∥22+d1∥Δx1∥22+2b1∫Ω‍x1|∇x1|2dx≤a1∥∇x1∥22+d12∥Δx1∥22+C1∥x1(x1+D0)∥∞m1∫Ω‍|x2|2dx.
This yields
(50)12ddt∥∇x1∥22≤C∥∇x1∥22+C1∥x2∥22.We now recall the following lemma.Lemma 13 (the uniform Gronwall lemma).
Letβ, ζ, and h be nonnegative functions in Lloc1[0,∞;R). Assume that β is absolutely continuous on (0,∞) and the following differential inequality is satisfied:
(51)dβdt≤ζβ+h,fort>0.
If there exists a finite time t1>0 and some q>0 such that
(52)∫tt+q‍ζ(τ)dτ≤A,∫tt+q‍β(τ)dτ≤B,∫tt+q‍h(τ)dτ≤C,
for any t>t1, where A,B, and C are some positive constants, then
(53)β(t)≤(Bq+C)eA,foranyt>t1+q.We apply the previous lemma to (50) with
(54)ζ(τ)=C,β(τ)=∥∇x1∥22,h(τ)=C1∥x2∥22,
and t1=t*, q=1, and we use the estimates via (31) and Lemma 12 to obtain
(55)∥∇x1(t+1)∥22≤C,t>t*.
The H1 estimates on the other components, x2 and x3, are obtained similarly.Recall that inℝ2 we have via the Sobolev embedding that H1(Ω)↪Lp(Ω),  for all p,
(56)limsupt→∞∥x1∥p≤C,
and similarly, for x2 and x3, we have
(57)limsupt→∞∥x2∥p≤C,limsupt→∞∥x3∥p≤C.Using the previous equation it is easy to show that the reaction termsg1, g2, and g3 are in Lp, for p>1. In general, p>N/2 is required [33], but in the current scenario N=2; thus the condition reduces to p>1. The estimates trivially follow from Hölder’s inequality, applied on the reaction terms along with (56) and (57). This proves the global existence result.


### 4.3. Existence of Global Attractor inL2(Ω)

We can now use the standard theory [32] to prove that the semigroup for (20)–(22) possesses a global attractor. Thus, we can state the following result.Theorem 14.
Consider the three-species food-chain model described by (20)–(22), 1<m3<m4, and 0<m2≤1. There exists a global attractor 𝒜 in H, for the solution semigroup {S(t)}t≥0 generated by this model.Proof.
We have shown that the system is well posed via Theorem6. Thus, there exists a well-defined semigroup {S(t)}t≥0:H→H. Lemma 12 establishes the existence of bounded absorbing sets in H. Thus, given a sequence {x1n(0)}n=1∞ that is bounded in L2(Ω), we know that, for t>t*,
(58)S(t)(x1n(0))⊂B⊂H1(Ω).
Here, B is the bounded absorbing set in H1(Ω). Now,  tn>t*, for any large enough n. Therefore, for tn, we have
(59)S(tn)(x1n(0))⊂B⊂H1(Ω).
This implies that we have the following uniform bound, via (57):
(60)∥S(tn)(x1n(0))∥H1(Ω)≤C1.
By standard functional analysis theory [32], a subsequence, still labelled as S(tn)(x1n(0)), exists such that
(61)S(tn)(x1n(0))⇀x1inH1(Ω),
which implies, via the compact Sobolev embedding of V↪H, that
(62)S(tn)(x1n(0))⟶x1inL2(Ω).
This yields the asymptotic compactness of the semigroup {S(t)}t≥0 in H. Similar analysis is possible for components x2 and x3. We have the existence of an absorbing set via Lemma 12 and the asymptotic compactness via (62). These in conjunction yield the existence of the global attractor, via standard methods [32].


## 4.1. Preliminaries

The system (20)–(22) is an example of a good semilinear parabolic system, and the standard theory for parabolic systems [31] ensures the existence of a nonnegative local solution. Next, comparison arguments can be used to derive global existence for initial data in L∞(Ω). We assume that x1, x2, and x3 are nonnegative because they are densities. This assumption ensures that x1+D0>0,  x1+D1>0,  x2+D2>0,  and x2+D3>0. The nonlinearities are thus well defined. Standard parabolic theory guarantees the uniform boundedness of x3. We state the following theorem.Theorem 6.
Consider the three-species food-chain model described by (20)–(22). For any positive initial data (x10,x20,x30)∈L2(Ω) and 1<m3<m4, 0<m2≤1, there exists a global classical solution (x1,x2,x3) to the system.Remark 7.
In this section, we demonstrate a stronger version of global existence. That is, we derive uniformL∞(Ω) bounds on the solution with the initial data only in L2(Ω).The opposite signed nonlinearities here cause problems in making directH1 estimates and proving the existence of bounded absorbing sets, for the x2 variable. To circumvent this difficulty, we resort to grouping estimates via appropriate addition of the equations at hand. This technique is very similar to what we adopted in [21]. This condition plays a key role in the dissipation process. If it is satisfied, there is often a global attractor. We will also show the existence of a global attractor for the reaction diffusion system (20)–(22).Remark 8.
Note, in [9] in order to proceed, we need to assume c≤w3/(h+v). Thus, our results in [9] are only true in the parameter regime, such that v<(w3/c)-D3. Furthermore in [9], strong L∞ boundedness requirements on the solutions are assumed. Here, we enforce no such requirement, and the methods of analysis are quite different from [9].Definition 9.
Consider a semigroup,S(t), acting on a reflexive Banach space, H. Then, the global attractor, 𝒜⊂H, for this semigroup is an object that has the following properties: (i)
𝒜 is compact in H,(ii)
𝒜 is invariant; that is, S(t)𝒜=𝒜,t≥0,(iii)
ifB is bounded in M, then distM(S(t)B,𝒜)→0, t→∞.
Next, various preliminaries are presented, detailing the phase spaces of interest and recalling certain standard theories. Let us define our phase spaces of interest as(24)H=[L2(Ω)]3,V=[H1(Ω)]3.
To prove the existence of a global attractor, we are required to show [32](i)
the existence of a bounded absorbing set in the phase space;(ii)
the asymptotic compactness property of the semigroup in question. These are defined next.Definition 10 (bounded absorbing set).
A bounded set,ℬ, in a reflexive Banach space, H, is called a bounded absorbing set if, for each bounded subset U of H, there is a time, T=T(U), such that S(t)U⊂ℬ for all t>T. The number T=T(U) is referred to as the compactification time for S(t)U. This is essentially the time after which the semigroup compactifies.Definition 11 (asymptotic compactness).
The semigroup{S(t)}t≥0:H→H associated with a dynamical system is said to be asymptotically compact in H if, for any {un(0)}n=1∞ bounded in H and a sequence of times {tn→∞},   S(tn)un(0) possesses a convergent subsequence in H.


## 4.2. Global Existence and Existence of Absorbing Sets inL2(Ω)

We begin by multiplying (20) by x1 and integrating by parts over Ω. This yields
(25)12ddt∥x1∥22=-d1∥∇x1∥22+a1∥x1∥22-b1∥x1∥33-w0∫Ω‍x1m1+1x2m2(x1+D0)m1dx.We then use Hölder’s inequality followed by Young’s inequality to obtain(26)12ddt∥x1∥22≤-d1∥∇x1∥22+b1∥x1∥33+C-b1∥x1∥33-w0∫Ω‍x1m1+1x2m2(x1+D0)m1dx.
Next, using the compact Sobolev embedding, H1(Ω)↪L2(Ω), and the positivity of x1 and x2, we find that
(27)ddt∥x1∥22+C1∥x1∥22≤C,
where C depends explicitly on b1 and a1.Thus, application of Gronwall’s Lemma gives us the following estimate:(28)∥x1∥22≤e-C1t∥x1(0)∥22+CC1.Therefore, there exists timet1 given explicitly by
(29)t1=max(0,ln(∥x1(0)∥22)C1),
such that, for all t≥t1, the following estimate holds uniformly:
(30)∥x1∥22≤1+C2C1.We now make a local in time estimate for∇x1. By integrating (26) in the time interval [t1,t1+1], we obtain
(31)∫t1t1+1‍∥∇x1∥22dt≤∥x1(t1)∥22+∫t1t1+1‍C2dt≤C.
Thus, via a mean value theorem for integrals, there exists a time t2∈[t1,t1+1] such that the following estimate holds:
(32)∥∇x1(t2)∥22≤C.We now move on to showing the existence of an absorbing set forx2 in L2(Ω). By multiplying (20) by w1 and (21) by w0, adding the two together, and setting w=w1x1+w0x2, we obtain
(33)∂w∂t=d2Δw+(d1-d2)w1Δx1+w1a1x1-w1b1x12-w0a2x2-w0w2(x2x1+D2)m2x3m2.
By multiplying (33) by w, integrating by parts over Ω, and using Hölder’s inequality followed by Young’s inequality, keeping in mind the positivity of the solutions, we find that
(34)12ddt∥w∥22+d2∥∇w∥22≤w1a1∫Ω‍x1wdx+∫Ω‍(d2-d1)w1∇x1·∇wdx≤C∥w∥22+C2∥x1∥22+C1∥∇x1∥22+d22∥∇w∥22.
Thus, by using the compact Sobolev embedding, H1(Ω)↪L2(Ω), we obtain
(35)ddt∥w∥22+C∥w∥22≤C1∥∇x1∥22+C2∥x1∥22.
Next, we multiply the previous equation by eCt and integrate in time from 0 to t to find
(36)∥w∥22≤e-Ct∥w(0)∥22+∫0t‍e-(t-τ)∥∇x1∥22dτ+C1(x10,t),
where C1(x10,t)=e-Ct∫0t‍e-C1τ∥x1(0)∥22dτ+(C/C1). This follows via (30).We now make the following estimate via time integration of (26), after multiplying through by eCt:
(37)∫0t‍eCτ∥∇x1∥22dτm≤CeCt-12∫0t‍(eτddτ∥x1∥22)m=CeCt-12[eCt∥x1(t)∥22-∥x10∥22-∫0t‍eCτ∥x1(τ)∥22dτ]m≤CeCt+∥x10∥22+∫0t‍eCτ∥x1(τ)∥22dτm≤CeCt+∥x10∥22.
This follows via (30).We now substitute the above into (36) and use the form of C1(x10,t) to obtain
(38)∥w∥22≤e-Ct∥w(0)∥22+C1(x10,t)+Ce-Ct(CeCt+∥x10∥22).
Therefore, there exists time t3, given explicitly by
(39)t3=max(0,ln(w1∥x1(0)∥22+w0∥x1(0)∥22)C),
such that, for all t≥t3, the following estimate holds uniformly:
(40)∥x2∥22≤1w0∥w∥22≤1+C+CC1.
Therefore,
(41)limsupt→∞∥x2∥22≤1w0limsupt→∞∥w∥22≤C,
where C is a constant that is independent of the time and the initial data.We next derive local in time estimates for∇x2. We proceed by multiplying (21) by x2 and integrating by parts over Ω to obtain
(42)12ddt∥x2∥22=-d2∥∇x2∥22-a2∥x2∥22+w1∫Ω‍(x1(x1+D0))m1x2m2+1dx-w2∫Ω‍x2m2+1(x2+D2)m2x3m2dx.We now use Hölder’s inequality, positivity of solutions, the embeddingL2↪Lm2+1 (via assumption on m2), and the earlier estimate via (40) to yield
(43)ddt∥x2∥22+d2∥∇x2∥22≤∥x1x1+D0∥∞m1∥x2∥22≤C1.
By integrating this inequality in the time interval [t,t+1], for t≥t3, we obtain
(44)∫tt+1‍∥∇x2∥22ds≤∥x2(t)∥22+∫t1t1+1‍C1|Ω|ds≤C.
The existence of an absorbing set for x3 in L2(Ω) follows easily via the form of (22).Standard estimates imply the existence of a timet5 given explicitly by
(45)t5=max(0,ln(|x3(0)|22)C),
such that, for all t≥t5, the following estimates hold uniformly:
(46)∥x3∥22≤1+C2C,∫tt+1‍∥∇x3∥22ds≤C.We can now state the following lemma.Lemma 12.
Consider(x1,x2,x3) which are solutions to the diffusive three-species food-chain model described via (20)–(22), with (x10,x20,x30)∈L2(Ω), 1<m3<m4, and  0<m2≤1. There exists a time  t*=max(t1,t3,t5) and a constant C independent of time and the initial data and dependent only on the parameters in (20)–(22), such that, for any t>t*, the following uniform estimates hold:
(47)∥x1∥22≤C,∥x2∥22≤C,∥x3∥22≤C.We now make theH1 estimate. We take the inner product of (20) with -Δx1 and integrate by parts over Ω. Thus, we obtain
(48)12ddt∥∇x1∥22+d1∥Δx1∥22+2b1∫Ω‍x1|∇x1|2dx=a1∥∇x1∥22+w0∫Ω‍(x1(x1+D0))m1(-Δx1)(x2)m2dx.
We apply the Cauchy-Schwartz inequality, the Cauchy with epsilon, and Young’s inequalities on the last term, along with the embedding L2↪L2m2 (via assumption on m2) to obtain
(49)12ddt∥∇x1∥22+d1∥Δx1∥22+2b1∫Ω‍x1|∇x1|2dx≤a1∥∇x1∥22+d12∥Δx1∥22+C1∥x1(x1+D0)∥∞m1∫Ω‍|x2|2dx.
This yields
(50)12ddt∥∇x1∥22≤C∥∇x1∥22+C1∥x2∥22.We now recall the following lemma.Lemma 13 (the uniform Gronwall lemma).
Letβ, ζ, and h be nonnegative functions in Lloc1[0,∞;R). Assume that β is absolutely continuous on (0,∞) and the following differential inequality is satisfied:
(51)dβdt≤ζβ+h,fort>0.
If there exists a finite time t1>0 and some q>0 such that
(52)∫tt+q‍ζ(τ)dτ≤A,∫tt+q‍β(τ)dτ≤B,∫tt+q‍h(τ)dτ≤C,
for any t>t1, where A,B, and C are some positive constants, then
(53)β(t)≤(Bq+C)eA,foranyt>t1+q.We apply the previous lemma to (50) with
(54)ζ(τ)=C,β(τ)=∥∇x1∥22,h(τ)=C1∥x2∥22,
and t1=t*, q=1, and we use the estimates via (31) and Lemma 12 to obtain
(55)∥∇x1(t+1)∥22≤C,t>t*.
The H1 estimates on the other components, x2 and x3, are obtained similarly.Recall that inℝ2 we have via the Sobolev embedding that H1(Ω)↪Lp(Ω),  for all p,
(56)limsupt→∞∥x1∥p≤C,
and similarly, for x2 and x3, we have
(57)limsupt→∞∥x2∥p≤C,limsupt→∞∥x3∥p≤C.Using the previous equation it is easy to show that the reaction termsg1, g2, and g3 are in Lp, for p>1. In general, p>N/2 is required [33], but in the current scenario N=2; thus the condition reduces to p>1. The estimates trivially follow from Hölder’s inequality, applied on the reaction terms along with (56) and (57). This proves the global existence result.


## 4.3. Existence of Global Attractor inL2(Ω)

We can now use the standard theory [32] to prove that the semigroup for (20)–(22) possesses a global attractor. Thus, we can state the following result.Theorem 14.
Consider the three-species food-chain model described by (20)–(22), 1<m3<m4, and 0<m2≤1. There exists a global attractor 𝒜 in H, for the solution semigroup {S(t)}t≥0 generated by this model.Proof.
We have shown that the system is well posed via Theorem6. Thus, there exists a well-defined semigroup {S(t)}t≥0:H→H. Lemma 12 establishes the existence of bounded absorbing sets in H. Thus, given a sequence {x1n(0)}n=1∞ that is bounded in L2(Ω), we know that, for t>t*,
(58)S(t)(x1n(0))⊂B⊂H1(Ω).
Here, B is the bounded absorbing set in H1(Ω). Now,  tn>t*, for any large enough n. Therefore, for tn, we have
(59)S(tn)(x1n(0))⊂B⊂H1(Ω).
This implies that we have the following uniform bound, via (57):
(60)∥S(tn)(x1n(0))∥H1(Ω)≤C1.
By standard functional analysis theory [32], a subsequence, still labelled as S(tn)(x1n(0)), exists such that
(61)S(tn)(x1n(0))⇀x1inH1(Ω),
which implies, via the compact Sobolev embedding of V↪H, that
(62)S(tn)(x1n(0))⟶x1inL2(Ω).
This yields the asymptotic compactness of the semigroup {S(t)}t≥0 in H. Similar analysis is possible for components x2 and x3. We have the existence of an absorbing set via Lemma 12 and the asymptotic compactness via (62). These in conjunction yield the existence of the global attractor, via standard methods [32].


## 5. Numerical Simulation and Attractor Reconstruction

We now carry out numerical simulations of (2)–(4). Our goal is to firstly show spatiotemporal chaos, even if m3<m4=2. This is indeed the case in certain parameter regime; see Figure 3. We also want to numerically validate the finite time blow-up results in 1D as well as in 2D. For 1D simulation, The system is simulated in MATLAB (R2010) via the PDEPE solver for partial differential equation. The algorithm essentially uses a central difference in space and an implicit time stepping method. All the simulations performed have been refined several times on spatial grids with 300, 600, 900, and 1200 points on the domain of length L=π. These refinements lead to the same general shape and structure of the figures. Table 1 lists the parameter values used in the simulations.Table 1
Parameters used in simulations.
Figure1
Figure2
Figure3
d
1
0.01
1
0.01
d
2
0.000001
1
0.000001
d
3
0.0001
1
0.0001
a
1
2
2
2
a
2
0.8
0.8
0.8
b
1
0.15
0.15
0.15
c
0.07
0.07
0.04
m
3
2
2
1.97
w
0
0.55
0.55
0.55
w
1
2
2
2
w
2
0.5
0.5
0.5
w
3
1.2
1.2
1.2
D
0
10
10
10
D
1
13
13
13
D
2
10
10
10
D
3
20
20
20In order to explore the dynamics of the model in 2D, we use a finite difference method. A forward difference scheme is used for the reaction terms. For the diffusion terms, a standard five-point explicit finite difference scheme is used. The numerical simulation is carried out at different time levels for two dimensional spatial model system. The system of equations is numerically solved over 200× 200 mesh points, on a domain of size Lx×Ly, with spatial resolution Δx = Δy = 1 and time step Δt=0.1. The initial condition used is a small perturbation about (2.1,2.9,1.9), and the boundary conditions used are the no flux Neumann conditions. Values for Lx and Ly are taken to be 200 each. Note this is fine, as the blow-up results hold for Neumann boundary conditions as well. See Figure 1 to visualise the blowup in 1D, and Figure 2 to visualise the blowup in 2D.Blowup is numerically demonstrated for the system (2)–(4), in these plots, when the mutual interference parameters are m1=1,m2=1,andm3=2. This is exactly the model proposed in [4]. We see blowup in the ODE model in the left, at t=24, and PDE model in the right, at t=41. Note here we take c=0.07>(w3/D3)=0.06.
(a)
(b)Figure 2
Here, finite time blowup for the model system in two dimensional spatial domain is seen. This occurs at timet=29.4.The densities of thex1andx2 species are shown as contour plots in the x-t-plane. The corresponding parameters are listed in Table 1. The long-time dynamics (we run up to time t=3000) is seen to be that of spatiotemporal chaos. This shows that one can have interesting dynamics even in the case that m3<m4, whilst avoiding finite time blowup.
(a)
(b)Our next goal is to quantify the chaotic dynamics present in the system. The dynamics of the three-species model described via (2)–(4) is depicted through the reconstruction of the trajectories in phase space with the aide of delay-time method. This method, proposed by Takens and Mane [34] ensures equivalence between the topological properties of actual and reconstructed attractors. Numerical simulations were conducted for three values of the mutual interference parameter m3. That is m3=2, m3=1.9, and m3=1.2. In all cases, m4 was set equal to 2. The system of PDE (2)–(4) was solved again via the MATLAB routine PDEPE. We then fixed certain points in space, and recorded the trajectory of the solution in time, yielding a time series. Very interesting behavior was observed; this consists of suppression of chaos. In fact for a value of m3=2, chaos is found when c=0.03 in Table 1 in Upadhyay et al. [35]; see Figure 4(a).The figures above show the dependence of the dynamics onm3. For m3=2, one sees chaotic dynamics with the largest Lyapunov exponent equal to 2.23. Reducing m3 to 1.9 brings the largest lyapunov exponent equal to 0.26. Further reducing it to 1.2 brings the dynamics to a limit cycle.
(a)
(b)
(c)We next calculated the dimension of this attractor, using a combination of singular-value decomposition and the Grassberger-Procaccia algorithm, [36]. We essentially want to embed the time series in an m dimensional embedding space of embedded vectors, which take the typical form
(63)ys=(x(t0+(s-1)l),x(t0+(s-1)l+τ),…,mx(t0+(s-1)l+(m-1)τ))T.Here,τ is called the “delay-time," l is the sampling interval, and w=(m-1)τ is the “window length" which represents the time spanned by each embedding vector. This algorithm entails choosing m and the delay, so that the window is sufficiently large. We then perform a singular value decomposition and calculate the correlation dimension, from the correlation integral. We continue to refine the embedding dimension till we have maximized the straight line parts in the log-log plot of the correlation integrals. The dimension of this attractor is approximately equal to 2.3 while the value of the largest Lyapunov exponent, estimated with the method proposed by Lai and Chen [37], is equal to 2.23. This clearly indicates that the dynamics is chaotic. Decreasing slightly the value of m3 to 1.9, the dynamics remained chaotic and the dimension of the attractor remained almost unchanged, that is, equal to 2.3 while the largest Lyapunov exponent had decreased to 0.26; see Figure 4(b). This tells us that the divergence of close trajectories becomes now very slow. The important point here is that the proposed modification to the Parshad-Upadhyay model and subsequently the classical Upadhyay-Rai models still exhibits chaos in certain parameter regime. That is for 1.9<m3<m4=2. Note since w0<w1, this parameter set is not very realistic biologically. However, when we decrease further the value of m3 to m3=1.2, the dynamics of the system changes radically and becomes oscillatory with single period; this is depicted by a limit cycle in phase space; see Figure 4(c). From this numerical analysis, it seems plausible that the mutual interference parameter m3 plays a crucial role in the dynamics of the three-species model. Setting a value of the parameter m3 (m3=1.2), very different from the classical one m3=2, suppresses the chaos predicted by the model when m3=2. However, keeping the parameters close to 2, that is 1.9<m3<2, still retains chaotic dynamics, whilst eliminating the possibility of finite time blowup.


## 6. Discussion

Various nonlinear effects, like structural instability, dissipative structures, dynamical chaos, and finite time blowup, do exist in food chain models. However, the problem of detecting these phenomena in real ecosystems is far from being well understood. Various studies have suggested that the biology of the top generalist predator has an important role to play as far as the dynamics of food-chain models is concerned [10, 38]. The nature of interactions between the populations and communities may be responsible for chaos evading its capture in the wild. In this work, we have shown that a large class of three-species food-chain models [4, 7, 9] can blowup in finite time, for certain parameter regime and large enough initial data. We reiterate that the situation can be worse than unbounded growth if c>w3/D3, as reported earlier [5]. We propose a possible correction based on the analysis of the way mating is modeled in (4). That is, if we restrict m3 to the range 1<m3<m4, then finite time blowup is prevented, and this is independent of the sign of c-w3/D3. However, if c-w3/D3<0 and x2=0, the top predator can still persist, for 1<m3<m4. Thus, this formulation plausibly models a true generalist predator x3, one that does not go extinct, even if its favorite food source does. Finally, by varying m3 in the range 1<m3<m4, one can adapt the new model to populations where the numbers of males and females vary. If the numbers are very close, we take m3=2-ϵ, ϵ≪1. One could also model a population where the males far outnumber the females or vice versa. Here one might take m3=1.5, saying we have roughly 10 females per 100 males in a population or vice versa. Thus, making the case for the model in structured populations, where such demographics may be dominant. Also, simulations performed by varying the mutual interference parameter m3 indicate clearly that there is a bifurcation point m3*, in the m3 parameter space, such that, chaotic dynamics is seen for m3>m3* and oscillatory dynamics for m3<m3*. It would be interesting to obtain the actual value of the mating parameter m3 from real field data. Comparing this “real value" to m3* might then be a means to again answer the long standing question of why chaos has evaded its capture in the wild. Lastly, we remark that the blow-up result obtained via Theorem 2 is sufficient but not necessary. It is not optimal in terms of data. That is, it might still be possible for blowup to occur if the data does not meet the largeness condition
(64)∫Ω‍x30(x)W1(x)dx>(λ1d31(c-w3/D3))1/(m3-1).For the classical diffusive Upadhyay-Rai model [4], we show blowup numerically; see Figure 1. Here, we have precisely all the parameters previously stated, including the first eigenvalue and eigenfunction of the Dirichlet laplacian. We see that, for the parameters chosen to demonstrate blowup in Figure 1,
(65)∫Ω‍x30(x)W1(x)dx=0.8<1.5=(λ1d31(c-w3/D3)).
Thus blowup takes place, even for data smaller than that required by the largeness condition. It would thus be very interesting to try and sharpen this largeness requirement on the data, perhaps via employing certain other techniques available in the blow-up literature, than the version of the first eigenvalue method that we have resorted to currently.



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*Source: 424062-2013-06-13.xml*