Introduction

Pollution has many adverse effects on ecosystems. In order to better understand the pollution problem, many works in the literature couple the dynamics of populations in ecosystems, with those of the pollutants therein (Hallam and De Luna 1984; Misra and Babu 2014, 2016). This serves two purposes

  • One can better understand the spread and subsequent control of these pollutants, so as to provide a cleaner living environment.

  • One can better understand the effects of the pollutants on the species residing in the ecosystem, so as to protect them, and maintain biodiversity.

In this regard the work of the authors in Misra and Babu (2016) is very interesting, and such investigations are highly desireable. Therein, two five-species ordinary differential equation (ODE) models are proposed and analyzed in an attempt to understand the effect of toxicity/pollution on a three species food chain, modeled via a combination of a modified Leslie-Gower type scheme and Holling type II functional response (Upadhyay and Rai 1997; Aziz-Alaoui 2002). However, one must take extreme care in deriving global existence results for such systems, given recent results that show finite time blow-up in such models (Parshad et al. 2013; Parshad 2015; Parshad and Basheer 2016).

The purpose of this short communication is to remark on certain results in Misra and Babu (2016). The authors therein derive global boundedness results for solutions to the ODE model systems proposed, under certain restrictions on the parameters. We note, these results are incorrect. In particular there is no bound on the solutions to the proposed models, even under the parametric restrictions imposed in Misra and Babu (2016). Our primary contributions in the current manuscript are:

  • We consider the second model proposed in Misra and Babu (2016), and show solutions to this model system are not bounded. In fact the solutions can blow-up in finite time, for sufficiently large initial data. This is shown via theorem 2.3.

  • Similar results are proved via corollary 1, for the first model system proposed in Misra and Babu (2016).

  • We demonstrate our results via numerical simulations. In particular, we choose parameters satisfying the parameter restrictions derived in Misra and Babu (2016), and show that solutions of the models considered, still blow-up in finite time, for sufficiently large initial condition.

  • We provide concluding remarks and discuss interesting future directions.

Finite time blow-up

Given a system of ODE’s, depending on the nonlinearities in the equations, one might not expect a solution to always exist globally in time. In particular, solutions of some ODE’s may blow-up in finite time (Straughan 1998). Recall,

Definition 2.1

(finite time blow-up) Given a ODE, with suitable initial conditions, one says finite time blow-up occurs if

$$\begin{aligned} \lim _{t\rightarrow T^{*}<\infty }\Vert z \Vert _{\infty } \rightarrow \infty , \end{aligned}$$

here the norm \(\Vert \cdot \Vert\) is the supremum norm, z is the solution to the ODE in question, and \(T^{*} < \infty\), is the blow-up time.

In the literature, finite time blow-up is also referred to as an explosive instability (Straughan 1998), and there are many interpretations of blow-up in physical phenomenon. The interested reader is referred to Quittner and Souplet (2007); Straughan (1998).

We now turn our attention back to Misra and Babu (2016). Of the two models proposed in Misra and Babu (2016), the non-dimensionalised form of the second one is,

$$\begin{aligned} \frac{dX}{dt} = a_{0}X - b_{0}X^2 - \left( \frac{v_{0}XY}{d_{0}+X}\right) , \end{aligned}$$
(1)
$$\begin{aligned} \frac{dY}{dt} = \left( \frac{v_{1}XY}{d_{1} + X}\right) - a_{1} Y - i_{2}YC_{0}- \left( \frac{v_{2}Y Z}{d_{2}+Y}\right) , \end{aligned}$$
(2)
$$\begin{aligned} \frac{dZ}{dt} = c_{3}Z^{2}-v_{3}\frac{Z^{2}}{d_{3}+Y}, \end{aligned}$$
(3)
$$\begin{aligned} \frac{dC_{0}}{dt} = c_{1}X C_{E} - s_{1}C_{0} - s_{2}XC_{0}, \end{aligned}$$
(4)
$$\begin{aligned} \frac{dC_{E}}{dt} = q_{0} - s_{3}C_{E} - c_{1}XC_{E} + s_{2}XC_{0}. \end{aligned}$$
(5)

Following the scaling from Misra and Babu (2016), one converts (1–5) into,

$$\begin{aligned} \frac{dx}{dt} = x(1-x) - \left( \frac{xy}{a+x}\right) , \end{aligned}$$
(6)
$$\begin{aligned} \frac{dy}{dt} = \left( \frac{cxy}{a + x}\right) - \beta _{2} y c_{0} - b y - \left( \frac{y z}{d+y}\right) , \end{aligned}$$
(7)
$$\begin{aligned} \frac{dz}{dt} = pz^{2}-q\frac{z^{2}}{r+y}, \end{aligned}$$
(8)
$$\begin{aligned} \frac{dc_{0}}{dt} = t_{2}x c_{e} - o_{1}c_{0} - t_{2}xc_{0}, \end{aligned}$$
(9)
$$\begin{aligned} \frac{dc_{e}}{dt} = p_{1} - o_{2}c_{e} - u_{2}xc_{e} + u_{2}xc_{o}. \end{aligned}$$
(10)

Remark 1

(1–5) is used for the purpose of numerical simulations in Misra and Babu (2016), while (6–10) is used for analysis. In keeping with this, we do the very same in the current manuscript.

For complete details on the above model formulation the reader is refereed to Upadhyay and Rai (1997), Aziz-Alaoui (2002), Misra and Babu (2016) and Leslie (1948). Essentially z is a specialist top predator, depredating on a specialist middle predator y. The interaction between z and y is modeled via a modified Leslie-Gower scheme. Also the middle predator y depredates on a prey species x. The interactions between y and x are modeled via a standard Holling type II functional response. Also \(c_{0}\) and \(c_{e}\) represent the pollutant concentration in the ecosystem, in both the prey x, and the environment respectively. There is an additional death term \(-\beta _{2} c_{0} y\), in the middle predator, due to toxicant concentration. This is realistic, as there could easily be enhanced death in the middle predator y, due to ingestion of toxic prey perse.

We first recall the following result (Theorem 4.1 from Misra and Babu (2016))

Theorem 2.2

Let us assume

$$\begin{aligned} c + \frac{c}{4b} + r < \frac{q}{p}, \end{aligned}$$
(11)

and let \(\Omega _{2}\) be the set defined by:

$$\begin{aligned}&\Omega _{2} = \bigg \{ (x,y,z,c_{0},c_{e}) \in \mathbb {R}^{5}_{+} : 0 \le x \le 1, 0 \le x + \frac{y}{c} \le 1 + \frac{1}{4b}, \nonumber \\&\quad x+\frac{y}{c} + \alpha z \le 1 + \frac{1}{4b} + \frac{M}{b}, c_{0} + c_{e} \le \frac{1+p_{0} t_{1}}{\theta _{2}} \bigg \} \nonumber \\ \end{aligned}$$
(12)

where \(\alpha = \frac{1}{b^2(c+\frac{c}{4b} + r)}\) , M = \(\frac{1}{4(q-(c+\frac{c}{4b} + r)p)}\) and \(\theta _{2} = \min {(o_{1}u_{1}, t_{1}o_{2})}\) then

  • (i) \(\Omega _{2}\) is positively invariant

  • (2i) All non-negative solutions (i.e. solutions initiating in \(\mathbb {R}^{5}_{+}\)) of (6)-(10) are uniformly bounded forward in time, (thus they exist for all positive times), they eventually enter the attracting set \(\Omega _{2}\).

  • (3i) System (6–10) is dissipative

We next show that (Theorem 4.1 (2i), (3i) Misra and Babu (2016)) is incorrect. In particular we state the following theorem

Theorem 2.3

Consider the three species food chain model (6–10). Even if

$$\begin{aligned} c + \frac{c}{4b} + r < \frac{q}{p}, \end{aligned}$$
(13)

z(t) blows up in finite time, that is

$$\begin{aligned} \mathop {\lim }_{t \rightarrow T^{*} < \infty } ||z(t)||_{\infty } \rightarrow \infty , \end{aligned}$$
(14)

as long as the initial data \((y_0, z_0)\) are large enough.

Remark 2

Note the above theorem shows, that contrary to what is claimed in Misra and Babu (2016), there is no bound for z.

Proof

Here we will show that the z solving (8), blows-up in finite time. Consider (6–10), with positive initial conditions. By integrating (8) we obtain

$$\begin{aligned} -\frac{1}{z}+\frac{1}{z_{0}}=pt- q\underset{0}{\overset{t}{\int }}\frac{ds}{y+r}, \end{aligned}$$

which gives

$$\begin{aligned} z=\frac{1}{\frac{1}{z_{0}}-pt+ q\underset{0}{\overset{t}{\int }} \frac{ds}{y+r}}. \end{aligned}$$

Our goal is to show that the function: \(t\rightarrow \psi \left( t\right) =\frac{1}{ z_{0}}-pt+q\underset{0}{\overset{t}{\int }}\frac{dt}{y+r}\) vanishes at a time \(T^{**}>0\). Then the solution z will blow-up in finite time, at \(t=T^{**}\).

Since the reaction terms in (6–10) are continuous functions, then the solutions (On their maximal interval of existence) are classical and continuous and

$$\begin{aligned} \underset{t\rightarrow 0}{\lim }\left( \frac{1}{t}\underset{0}{\overset{t}{ \int }}\frac{ds}{y+r}\right) =\frac{1}{y_{0}+r}. \end{aligned}$$

For \(y_{0}\) chosen sufficiently large, there exists a \(\delta >0\) such that

$$\begin{aligned} \frac{1}{t}\underset{0}{\overset{t}{\int }}\frac{ds}{y+r}<\frac{p}{ 2q},\quad \text {\ for \, all }\, \it{t}\in ( 0,\delta ) . \end{aligned}$$

Then

$$\begin{aligned} \frac{1}{z_{0}}-pt + q\underset{0}{\overset{t}{\int }}\frac{ds}{ y+r}=\frac{1}{z_{0}}+\left[ -p+\frac{q}{t}\underset{0}{ \overset{t}{\int }}\frac{ds}{y+r}\right] t <\frac{1}{z_{0}}-\frac{p}{2 }t,\quad \text {for \, all }\, \it{t}\in ( 0,\delta ) . \end{aligned}$$

If now \(z_{0}\) is chosen sufficiently large, then we can find \(T^{*}\in ( 0,\delta )\) such that

$$\begin{aligned} \frac{1}{z_{0}}-\frac{p}{2}T^{*}=0. \end{aligned}$$

This entails

$$\begin{aligned} \psi \left( T^*\right) = \frac{1}{z_{0}}-pT^* + q\underset{0}{\overset{T^*}{\int }}\frac{ds}{y+r} < \frac{1}{z_{0}}-\frac{p}{2}T^{*} = 0. \end{aligned}$$
(15)

Thus one has \(\psi \left( T^*\right) < 0\), but \(\psi \left( 0\right) > 0\), and since \(\psi (t)\) is continuous in time, by application of the mean value theorem, we obtain the existence of some \(T^{**}\in ( 0,\delta)\), \(T^{**} < T^*\), s.t \(\psi \left( T^{**}\right) = 0\). This implies z(t) the solution of (8), blows-up in finite time, at \(t=T^{**}\). This proves the theorem. \(\square\)

We next state the following Theorem

Theorem 2.4

The model system (6–10), even under condition (13) is not dissipative in all of \(\mathbb {R}^{5}_{+}\) . It is dissipative only for initial data starting within \(\Omega _{2}\).

Proof

Via Theorem 2.3, there exists initial data in \(\mathbb {R}^{5}_{+}\), for which solutions blow-up in finite time, and thus do not enter the set \(\Omega _{2}\), which is claimed to be invariant and attracting. Thus system (6–10) is not dissipative. However for initial data initiated in \(\Omega _{2}\), the trajectories remain in \(\Omega _{2}\). \(\square\)

Remark 3

Note, the set \(\Omega _{2}\) is thus only invariant, not globally attracting for \(\mathbb {R}^{5}_{+}\).

We now consider the first model proposed in Misra and Babu (2016),

$$\begin{aligned} \frac{dx}{dt} = x(1-x)- \beta _{1}x c_{0} - \left( \frac{xy}{a+x}\right) , \end{aligned}$$
(16)
$$\begin{aligned} \frac{dy}{dt} = \left( \frac{cxy}{a + x}\right) - b y - \left( \frac{y z}{d+y}\right) , \end{aligned}$$
(17)
$$\begin{aligned} \frac{dz}{dt} = pz^{2}-q\frac{z^{2}}{r+y}, \end{aligned}$$
(18)
$$\begin{aligned} \frac{dc_{0}}{dt} = t_{1}x c_{e} - o_{1}c_{0} - t_{1}xc_{0}, \end{aligned}$$
(19)
$$\begin{aligned} \frac{dc_{e}}{dt} = p_{1} - o_{2}c_{e} - u_{1}xc_{e} + u_{1}xc_{o}. \end{aligned}$$
(20)

The dynamics here are essentially the same as that of the second model, except that now the prey x has an additional non-linear death term \(- \beta _{1}x c_{0}\), due to pollution/toxicants.

We next show that (Theorem 3.1 (2i), (3i) Misra and Babu (2016)) which claims global boundedness of the above model (16–20) is incorrect. In particular we state the following corollary

Corollary 1

Consider the three species food chain model (16–20). Even if

$$\begin{aligned} c + \frac{c}{4b} + r < \frac{q}{p}, \end{aligned}$$
(21)

z(t) blows up in finite time, that is

$$\begin{aligned} \mathop {\lim }_{t \rightarrow T^{*} < \infty } ||z(t)||_{\infty } \rightarrow \infty , \end{aligned}$$
(22)

as long as the initial data \((y_0, z_0)\) are large enough.

Proof

The proof follows by a direct application of theorem 2.3. That is we mimic the proof of theorem 2.3, by integrating (18) in time, and proceeding as earlier. Hence the result follows as a corollary. \(\square\)

Numerical simulation

In this section we perform simple numerical simulations that disprove theorem 4.1 (Misra and Babu 2016) (recapped in the current manuscript as theorem 2.2).

The following is the list of parameters given in Misra and Babu (2016), for numerical simulations around the interior equilibrium \((X^*,Y^*,Z^*,C^{*}_{0},C^{*}_{E})\) for (1–5): \(a_0=0.97, b_0=0.06, v_0=1, d_0=10, a_1=1, v_1=2, d_1=10, v_2=0.405, d_2=10, c_3=0.0403, v_3=1, d_3=20, i_2=0.85, c_1 = 0.0183, s_1 = 3.8199, s_2 = 0.0270, q_0 = 1.6883, s_3 = 0.6984.\)

However, for the above values using the scaling introduced in Misra and Babu (2016), we obtain

$$\begin{aligned} c+\frac{c}{4b} + r = 3.8372 > 1.5842 = \frac{q}{p} \end{aligned}$$
(23)

Remark 4

Note that theorem 4.1 from Misra and Babu (2016) (recapped in the current manuscript as theorem 2.2) requires that we strictly maintain

$$\begin{aligned} c+\frac{c}{4b} + r < \frac{q}{p} \end{aligned}$$
(24)

Hence showing blow-up numerically, for the above parameter set does not contradict the theorems in Misra and Babu (2016).

We proceed by changing \(v_{1}\). We set \(v_{1}=.23\), and keep all other parameters the same as above. With \(v_{1}=.23\) one easily calculates,

$$\begin{aligned} c+\frac{c}{4b} + r = 1.57 < 1.5842 = \frac{q}{p} \end{aligned}$$
(25)

Thus the parametric restrictions imposed via theorem 4.1 from Misra and Babu (2016) are satisfied, and so according to Misra and Babu (2016), we should have globally bounded solutions, for any initial condition in \(\mathbb {R}^{5}_{+}\). The following numerical simulation, see Fig. 1, shows this is not true.

Fig. 1
figure 1

In this simulation we consider the list of parameters mentioned earlier, but with \(v_{1}=.23\). We take large initial data for the middle predator Y, and top predator Z. The initial data tried is \((12.62, 400, 400, 0.103, 0.1741) \in \mathbb {R}^{5}_{+}\). For this set of parameters we have, \(c+\frac{c}{4b} + r = 1.57 < 1.5842 = \frac{q}{p}\). However, we still observe that Z blows-up at approximately \(t=0.053\). We show the solution just before the blow-up time, at \(t=0.052\). Hence there is no bound on the solutions for sufficiently large initial data

Full size image

Conclusion

In the current manuscript we have shown that the five-species models introduced in Misra and Babu (2016), do not posess bounded solutions, even under the parametric restrictions imposed in Misra and Babu (2016). Note, solutions to these models can actually blow-up in finite time. In summary,

  • We caution against deriving global boundedness results whence modeling ecosystem dynamics, via the modified Leslie-Gower scheme, introduced in Upadhyay and Rai (1997).

  • Any global boundedness results, where the modified Leslie-Gower scheme is incorporated, must impose restrictions on initial conditions, and specify them explicitly, such as in Parshad et al. (2014).

  • Possible future avenues of research should be aimed at introducing damping mechanisms, that can impede or delay the blow-up, such as Parshad et al. (2016). To this end, the additional non-linear death term in the middle predator \(-\beta _{2}y c_{0}\), due to ingestion of toxic prey, is a right step.

One can investigate the effect on blow-up times, or blow-up impedence altogether (for certain initial conditions), with and without such a non-linear term. This would make an extremely interesting future work.