Stability of the logistic population model with generalized piecewise constant delays
- Duygu Aruğaslan^{1}Email author and
- Leyla Güzel^{1}
https://doi.org/10.1186/s13662-015-0521-8
© Aruğaslan and Güzel 2015
Received: 8 January 2015
Accepted: 27 May 2015
Published: 10 June 2015
Abstract
In this paper, we consider the logistic equation with piecewise constant argument of generalized type. We analyze the stability of the trivial fixed point and the positive fixed point after reducing the equation into a nonautonomous difference equation. We also discuss the existence of bounded solutions for the reduced nonautonomous difference equation. Then we investigate the stability of the positive fixed point by means of Lyapunov’s second method developed for nonautonomous difference equations. We find conditions formulated through the parameters of the model and the argument function. We also present numerical simulations to validate our findings.
Keywords
MSC
1 Introduction and preliminaries
Theory of ordinary differential equations plays an important role for solving fundamental problems in population dynamics. However, in some cases, it may not provide consistent results that meet the realities. This fact may arise from the negligence of time delays, impacts or other specific phenomena that are not suitable to be modeled by classical differential equations.
Since the discrete moments of time where the argument changes its constancy may not be equally distanced, differential equations with piecewise constant argument of generalized type may lead to nonautonomous difference equations. To the best of our knowledge, it is the first time in the literature that a differential equation with piecewise constant argument of generalized type is reduced into a nonautonomous difference equation.
2 Preliminaries
Let us give the necessary definitions and theorems [12, 18, 19] that will be useful in the next section. Stability definitions for difference equations are similar to the ones given for classical ordinary differential equations. Concepts of stability and their definitions for difference equations can be found in the book [18].
Let the function f in (6) satisfy \(f(k,0)=0\) for all \(k\in \mathbb{N}\). Then (6) admits the trivial solution \(u=0\).
Definition 2.1
- (i)
\(x(t)\) is continuous on \(\mathbb{R}^{+}\);
- (ii)
the derivative \(x'(t)\) exists for \(t\in\mathbb{R}^{+}\) with the possible exception of the points \(\theta_{k}\), \(k\in\mathbb{N}\), where one-sided derivatives exist;
- (iii)
(5) is satisfied by \(x(t)\) on each interval \((\theta_{k},\theta_{k+1})\), \(k\in\mathbb{N}\), and it holds for the right derivative of \(x(t)\) at the points \(\theta_{k}\), \(k\in\mathbb{N}\).
Definition 2.2
A function \(\phi(r)\) is said to belong to the class \(\mathcal{K}\) if and only if \(\phi\in C[[0,\rho),\mathbb{R}^{+}]\), \(\phi (0)=0\) and \(\phi(r)\) is strictly increasing in r.
Define \(S_{\rho}=\{u\in\mathbb{R}^{n}:\Vert u\Vert <\rho\}\).
Definition 2.3
- (i)
\(w(0)=0\) and
- (ii)
\(w(u)>0\) for all \(u\neq0\), \(u\in\mathbb{S}_{\rho}\).
Definition 2.4
- (i)
\(V(k,0)=0\) for all \(k\in\mathbb{N}\) and
- (ii)
\(V(k,u)\geq w(u)\) for all \(k\in\mathbb{N}\) and for all \(u\in S_{\rho}\).
Theorem 2.5
If there exists a positive definite scalar function \(V(k,u)\in C[\mathbb{N}\times S_{\rho},\mathbb{R}^{+}]\) such that \(\Delta V_{\text{(6)}}(k,u(k))\leq0\), then the trivial solution \(u=0\) of the difference equation (6) is stable.
Theorem 2.6
If there exists a positive definite scalar function \(V(k,u)\in C[\mathbb{N}\times S_{\rho},\mathbb{R}^{+}]\) such that \(\Delta V_{\text{(6)}}(k,u(k))\leq-\alpha(\Vert u(k)\Vert )\), where \(\alpha\in \mathcal{K}\), then the trivial solution \(u=0\) of the difference equation (6) is asymptotically stable.
3 Main results
- (A1)
there exists a positive constant θ such that \(\theta _{k+1}-\theta_{k}=\theta\) for all \(k\in\mathbb{N}\).
Theorem 3.1
If \(a<0\) then the fixed point \(u=0\) of (9) is asymptotically stable and it is unstable if \(a>0\). If \(0< a\theta<2\) then the fixed point \(u=\frac{a}{b}\) of (9) is asymptotically stable, and it is unstable if \(a<0\) or \(a\theta>2\).
Proof
For the fixed point \(u=0\), \(f'(0)={\mathrm{e}}^{a\theta }<1\) if and only if \(a<0\). For the fixed point \(u=\frac{a}{b}\), the conclusion follows from the equality \(f'(\frac{a}{b})=1-a\theta\). It is clear that \(|1-a\theta|<1\) if and only if \(0< a\theta<2\). Thus, the proof is completed. □
- (A2)
There exist positive constants \(\underline{\theta}\) and \(\overline{\theta}\) such that \(\underline{\theta}\leq\theta_{k+1}-\theta _{k}=g(k)\leq\overline{\theta}\) for all \(k\in\mathbb{N}\).
Let \(u(k)\) be any solution of (10) with \(u(0)=u_{0}>0\) for biological reasons. From now on, we shall investigate the stability of the fixed points \(u=0\) and \(u=\frac{a}{b}\) of the nonautonomous difference equation (10). We shall study the stability of the positive fixed point \(u=\frac{a}{b}\) by means of Lyapunov’s second method.
Theorem 3.2
If \(a>0\) then the trivial solution \(u=0\) of the difference equation (10) is unstable.
Proof
Theorem 3.3
If \(a<0\) then the trivial solution \(u=0\) of the difference equation (10) is asymptotically stable.
Proof
Assume that \(a<0\). Let \(\varepsilon>0\) be given. Consider a solution \(u(k)\), \(k\in\mathbb{N}\), of (10) with \(u(0)=u_{0}>0\). Choose \(\delta=\min\{\varepsilon,\frac{a}{b}\}\). For \(0< u_{0}<\delta\), we can see that \(u(k)\), \(k\in\mathbb{N}\), is a strictly decreasing sequence. Thus \(u(k)< u_{0}<\delta\leq\varepsilon\) for all \(k=1,2,\ldots\) . This shows that \(u=0\) is stable. For the same δ, it can be shown easily that \(\lim_{k\rightarrow\infty }u(k)=0\). Therefore, the trivial solution \(u=0\) of the difference equation (10) is asymptotically stable. □
Theorem 3.4
If \(a<0\) then \(u=\frac{a}{b}\) of the difference equation (10) is unstable.
Proof
We take \(a<0\) and suppose the contrary. Let \(u(k)\), \(k\in\mathbb{N}\), denote the solution of (10) with \(u(0)=u_{0}>0\). Then for \(\varepsilon=\frac{a}{2b}\), we can find a \(\delta>0\) such that \(\vert u_{0}-\frac{a}{b}\vert <\delta\) implies \(\vert u(k)-\frac {a}{b}\vert <\frac{a}{2b}\) for all \(k\in\mathbb{N}\). For \(\frac {a}{b}< u_{0}<\frac{a}{b}+\delta\), \(u(k)\), \(k\in\mathbb{N}\), is a strictly increasing sequence bounded above by \(\frac{3a}{2b}\). Thus, the solution converges to a number \(L\in(\frac{a}{b},\frac{3a}{2b}]\). However, when we take the limit of the inequality \(u(k+1)\geq u(k){\mathrm {e}}^{(a-bu(k))\underline{\theta}}\) as \(k\rightarrow\infty\), it gives the contradiction \(L\leq\frac{a}{b}\). For \(\frac{a}{b}-\delta< u_{0}<\frac {a}{b}\), \(u(k)\), \(k\in\mathbb{N}\), is a strictly decreasing sequence bounded below by \(\frac{a}{2b}\). Then \(\frac{a}{2b}\leq L=\lim_{k\rightarrow\infty}u(k)<\frac{a}{b}\). Letting \(k\rightarrow\infty\) in the inequality \(u(k+1)\leq u(k){\mathrm{e}}^{(a-bu(k))\underline{\theta}}\), we find \(L\geq\frac{a}{b}\), a contradiction. As a consequence, \(u=\frac {a}{b}\) of the difference equation (10) is unstable. □
Now, we prove the existence of bounded solutions for (10), which will be needed in Theorem 3.9.
Definition 3.5
A set \(\mathcal{V}\subset\mathbb{R}\) is called a positively invariant set of (10) if \(u_{0}\in\mathcal{V}\) implies \(u(k)=u(k,u_{0})\in\mathcal{V}\) for all \(k\in\mathbb{N}\).
Lemma 3.6
If \(0< a\overline{\theta}\leq1\), then the set \(\Gamma=\{u\in\mathbb{R} : 0< u<\frac{a}{b}\}\) is positively invariant for the difference equation (10).
Proof
Let \(0< a\overline{\theta}\leq1\) and \(u(k)\), \(k\in \mathbb{N}\), be the solution of (10) starting at \(u(0)=u_{0}\in\Gamma \). Assume that \(0< u(m)<\frac{a}{b}\) for some \(m\in\mathbb{N}\). We have \(u(m)< u(m+1)=u(m){\mathrm{e}}^{(a-bu(m))(\theta_{m+1}-\theta_{m})}\leq u(m){\mathrm{e}}^{(a-bu(m))\overline{\theta}}\). Define \(H(u(m))=u(m){\mathrm {e}}^{(a-bu(m))\overline{\theta}}\). Since \(0< a\overline{\theta}\leq1\), we see that \(H(u(m))<\frac{a}{b}\) on the interval \((0,\frac{a}{b})\). Thus, we get \(u(m)< u(m+1)<\frac{a}{b}\), which implies that \(u(m+1)\in \Gamma\). It is seen by induction that \(u(k)\in\Gamma\) for all \(k\in \mathbb{N}\). □
Lemma 3.7
If \(0< a\overline{\theta}\leq1\) then the set \(\Gamma=\{u\in\mathbb{R} : 0< u\leq\kappa, \kappa\geq\frac {a}{b}\}\) is positively invariant for the difference equation (10).
Proof
Let \(0< a\overline{\theta}\leq1\) and \(u(k)\), \(k\in \mathbb{N}\), be the solution of (10) starting at \(u(0)=u_{0}\in\Gamma \). If \(u_{0}=\frac{a}{b}\in\Gamma\) then \(u(k)=\frac{a}{b}\in\Gamma\) for all \(k\in\mathbb{N}\). If \(u_{0}\in(0,\frac{a}{b})\subset\Gamma\), then we know from Lemma 3.6 that \(u(k)\in(0,\frac{a}{b})\subset \Gamma\) for all \(k\in\mathbb{N}\). Let us consider the case \(\frac {a}{b}< u_{0}\leq\kappa\). Then either \(\frac{a}{b}< u(k)\leq u_{0}\leq\kappa\) for all \(k\in\mathbb{N}\), or \(0< u(m)\leq\frac{a}{b}\) for some natural number \(m\geq1\), which implies that \(0< u(m)\leq u(k)\leq\frac{a}{b}\leq \kappa\) for all \(k\geq m\). In any case, we see that the conclusion is true. Thus, the proof is completed. □
Lemma 3.8
If \(a\overline{\theta}\geq1\) then the set \(\Gamma=\{u\in\mathbb{R} : 0< u\leq\kappa, \kappa\geq\frac {1}{b\overline{\theta}}{\mathrm{e}}^{a\overline{\theta}-1}\}\) is positively invariant for the difference equation (10).
Proof
Let \(a\overline{\theta}\geq1\) and \(u(k)\), \(k\in \mathbb{N}\), be the solution of (10) starting at \(u(0)=u_{0}\in\Gamma \). Assume that \(u(m)\in\Gamma\) for some \(m\in\mathbb{N}\). If \(0< u(m)\leq \frac{a}{b}\), then \(0< u(m)\leq u(m+1)\leq H(u(m))=u(m){\mathrm {e}}^{(a-bu(m))\overline{\theta}}\leq\frac{1}{b\overline{\theta}}{\mathrm {e}}^{a\overline{\theta}-1}\leq\kappa\). If \(\frac{a}{b}\leq u(m)\leq \kappa\), then \(0< u(m+1)\leq u(m)\leq\kappa\). Thus, \(u(k)\in\Gamma\) for all \(k\in\mathbb{N}\) by induction. □
If we combine the results of Lemma 3.7 and Lemma 3.8, we can state the next result.
Corollary 1
If \(a>0\) then the set \(\Gamma=\{u\in \mathbb{R} : 0< u\leq\kappa, \kappa\geq\frac{1}{b\overline{\theta }}{\mathrm{e}}^{a\overline{\theta}-1}\}\) is positively invariant for the difference equation (10).
Theorem 3.9
If \(a>0\), \(0< u_{0}\leq\kappa\), where \(\kappa >\frac{1}{b\overline{\theta}}{\mathrm{e}}^{a\overline{\theta}-1}\) and \(b\overline{\theta}^{2}(\kappa-\frac{a}{b})<2\underline{\theta}(\ln\kappa -\ln\frac{a}{b})\), then the fixed point \(u=\frac{a}{b}\) of the difference equation (10) is asymptotically stable.
Proof
The assumption \(b\overline{\theta}^{2}(\kappa-\frac{a}{b})<2\underline {\theta}(\ln\kappa-\ln\frac{a}{b})\) implies that \(a\overline{\theta }^{2}<2\underline{\theta}\). Hence, we get \(\Delta V_{\text{(11)}}(k,y)\leq-w(y)\), where \(w(y)=\frac{b^{2}}{a}(2\underline{\theta }-a\overline{\theta}^{2})y^{2}\) is a positive definite function. Then we conclude that \(\Delta V_{\text{(11)}}(k,y)\) is negative definite.
For \(0\leq y\leq\kappa-\frac{a}{b}\), consider the function \(F(y)=-2b\underline{\theta}\ln\frac{y+\frac{a}{b}}{\frac {a}{b}}+b^{2}\overline{\theta}^{2}y\). It can easily be seen that \(F(y)<0\), \(y\neq0\), whenever \(b\overline{\theta}^{2}(\kappa-\frac{a}{b})<2\underline {\theta}(\ln\kappa-\ln\frac{a}{b})\) and \(F(0)=0\). Since \(F(y)\) is negative definite, we derive that \(\Delta V_{\text{(11)}}(k,y)\) is negative definite.
Consequently, \(\Delta V_{\text{(11)}}(k,y)\) is negative definite independent of the sign of y. Then, according to Theorem 2.6, the fixed point \(y=0\) of the difference equation (11) is asymptotically stable. Thus, the fixed point \(u=\frac{a}{b}\) of the difference equation (10) is asymptotically stable. □
4 Conclusion
We see that a differential equation with piecewise constant argument, whose distance between the two consecutive switching moments is equal, can be reduced into an autonomous difference equation. However, if we have a differential equation with generalized piecewise constant argument whose switching moments are ordered arbitrarily, then it generates a nonautonomous difference equation. This fact stimulates us to study the effects of generalized piecewise constant arguments on the stability of the fixed points of the logistic equation. Our results show that the existence of a generalized piecewise constant argument influences the behavior of the solutions. As far as we know, it is the first time in the literature that one reduces a differential equation with piecewise constant argument of generalized type into a nonautonomous difference equation. This idea can be used for the investigation of differential equations with piecewise constant argument of generalized type.
Declarations
Acknowledgements
The authors would like to express their sincere gratitude to the referees for their constructive comments and suggestions which have significantly improved the paper. This work was supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK), TÜBİTAK 2209-A.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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