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Oscillation and Stability of Delay Models in Biology

Authors:
Ravi Agarwal at Texas A&M University – Kingsville
Ravi Agarwal
Donal O’Regan
Donal O’Regan
Samir H Saker at New Mansoura University
Samir H Saker
  • New Mansoura University
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Abstract

Environmental variation plays an important role in many biological and ecological dynamical systems. This monograph focuses on the study of oscillation and the stability of delay models occurring in biology. The book presents recent research results on the qualitative behavior of mathematical models under different physical and environmental conditions, covering dynamics including the distribution and consumption of food. Researchers in the fields of mathematical modeling, mathematical biology, and population dynamics will be particularly interested in this material. © 2014 Springer International Publishing Switzerland. All rights are reserved.
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1
ISBN 978-3-319-06556-4
Ravi P.Agarwal· DonalO'Regan
Samir H.Saker
Oscillation and
Stability of
Delay Models
in Biology
Oscillation and Stability of Delay
Models in Biology
Agarwal · O'Regan · Saker
Ravi P.Agarwal· DonalO'Regan· Samir H.Saker
Oscillation and Stability of Delay Models in Biology
Environmental variation plays an important role in many biological and ecological
dynamical systems. This monograph focuses on the study of oscillation and the
stability of delay models occurring in biology. The book presents recent research
results on the qualitative behavior of mathematical models under dierent physical
and environmental conditions, covering dynamics including the distribution and
consumption of food. Researchers in the elds of mathematical modeling, mathematical
biology, and population dynamics will be particularly interested in this material.
Mathematics
9 783319 065564

Chapters (6)

Logistic Models
Chapter
  • May 2014
  • Oscillation and Stability of Delay Models in Biology
  • pp.1-7
All processes in organisms, from the interaction of molecules to complex functions of the brain and other organs, obey physical laws. Mathematical modeling is an important step towards uncovering the organizational principles and dynamic behavior of biological systems.
Oscillation of Delay Logistic Models
Chapter
  • May 2014
  • Oscillation and Stability of Delay Models in Biology
  • pp.9-77
The qualitative study of mathematical models is important in applied mathematics, physics, meteorology, engineering, and population dynamics. In this chapter, we are concerned with the oscillation of solutions of different types of delay logistic models about their positive steady states.
Stability of Delay Logistic Models
Chapter
  • May 2014
  • Oscillation and Stability of Delay Models in Biology
  • pp.79-126
The stability of the equilibrium points is important in the study of mathematical models. The equilibrium point \(\overline{N}\) is locally stable if the solution of the model N(t) approaches \(\overline{N}\) as time increases for all the initial values, in some neighborhood of \(\overline{N}\).
Logistic Models with Piecewise Arguments
Chapter
  • May 2014
  • Oscillation and Stability of Delay Models in Biology
  • pp.127-213
Differential equation with piecewise continuous argument (or DEPCA) will be discussed in this chapter.
Food-Limited Population Models
Chapter
  • May 2014
  • Oscillation and Stability of Delay Models in Biology
  • pp.215-291
Smith [66] reasoned that a food-limited population in its growing stage requires food for both maintenance and growth, whereas, when the population has reached saturation level, food is needed for maintenance only. On the basis of these assumptions, Smith derived a model of the form $$\displaystyle{ \frac{dN(t)} {dt} = rN(t) \frac{K - N(t)} {K + crN(t)} }$$ (5.1) which is called the “food limited” population . Here N, r, and K are the mass of the population, the rate of increase with unlimited food, and the value of N at saturation, respectively. The constant 1∕c is the rate of replacement of mass in the population at saturation. Since a realistic model must include some of the past history of the population, Gopalsamy, Kulenovic and Ladas introduced the delay in (5.1) and considered the equation $$\displaystyle{ \frac{dN(t)} {dt} = rN(t) \frac{K - N(t-\tau )} {K + crN(t-\tau )}, }$$ as the delay “food-limited” population model, where r, K, c, and τ are positive constants.
Logistic Models with Diffusions
Chapter
  • May 2014
  • Oscillation and Stability of Delay Models in Biology
  • pp.293-334
Population dispersal plays an important role in the population dynamics which arises from environmental and ecological gradients in the habitat. We assume that the systems under consideration are allowed to diffuse spatially besides evolving in time. The spatial diffusion arises from the tendency of species to migrate towards regions of lower population density where the life is better. The most familiar model systems incorporating these features are reaction diffusion equations.
... Therefore, we decide to introduce time delay into the ecological system cooperating to the delay feature of morphogens. Since time delay can cause disturbance, loss of stability of the equilibrium state and emergence of bifurcation phenomenon in the system, that can bring rich dynamical behaviors (see [1,2,7,8,11,13,15,[18][19][20][21]). In this paper, for our particular interests, we add the distributed delay into system (1.2) and investigate the rich dynamics of the system. ...
Bifurcations and dynamical behaviors for a generalized Maginu model with distributed delay
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This paper is committed to study the dynamical behaviors of a generalized reaction-diffusion Maginu model with distributed delay. We investigate the stability of the positive equilibrium and the existence of periodic solutions bifurcating from the positive equilibrium. Further, by using the center manifold theorem and the normal form theory, we derive the precise conditions to judge the bifurcation direction and the stability of the bifurcating periodic solutions. Importantly, we deduce the exact conditions in terms of the diffusion coefficients to guarantee the occurrence of Turing instability for both homogeneous equilibrium solution and the Hopf bifurcating periodic solutions. Intuitively, numerical simulations are used to support our theoretical analysis.
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... In the last few decades, as a significant branch of FDEs, impulsive differential equation (IDE, for short), which provides a natural description of observed evolution processes, has been emerging as a very meaningful research area. In addition, IDEs are also as important mathematical tools for better understanding real-world problems (see, for instance, [4][5][6][7][8]). Hence, many authors have used IDEs to describe some phenomenon with abrupt changes, such as, harvest, disease, control theory of dynamical systems and so on. ...
Multiple solutions for a class of BVPs of fractional discontinuous differential equations with impulses
Article
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  • Jan 2023
In this paper, we mainly study the following boundary value problems of fractional discontinuous differential equations with impulses: \begin{document}{tCD0+RΛ(t)=E(t)ϝ(t,Λ(t)), a.e. t∈Q,△Λ∣t=tκ=Φκ(Λ(tκ)), κ=1, 2, ⋯ , m,△Λ′∣t=tκ=0, κ=1, 2, ⋯ , m,ϑΛ(0)−χΛ(1)=∫01ϱ1(υ)Λ(υ)dυ,ζΛ′(0)−δΛ′(1)=∫01ϱ2(υ)Λ(υ)dυ, \hskip 3mm \left\{ \begin{array}{lll} _{t}^{C} \mathcal {D}^{\mathfrak{R}}_{0^{+}}\Lambda(t) = \mathcal {E}(t)\digamma(t, \Lambda(t)), \ a.e.\ t\in Q, \\ \triangle \Lambda|_{t = t_{{\kappa}}} = \Phi_{{\kappa}}(\Lambda(t_{{\kappa}})), \ {\kappa} = 1, \ 2, \ \cdots, \ m, \\ \triangle \Lambda'|_{t = t_{{\kappa}}} = 0, \ {\kappa} = 1, \ 2, \ \cdots, \ m, \\ {\vartheta} \Lambda(0)-{\chi} \Lambda(1) = \int_{0}^{1}\varrho_{1}({\upsilon})\Lambda({\upsilon})d{\upsilon}, \\ {\zeta} \Lambda'(0)-\delta \Lambda'(1) = \int_{0}^{1}\varrho_{2}({\upsilon})\Lambda({\upsilon})d{\upsilon}, \end{array}\right. \end{document} where ϑ>χ>0, ζ>δ>0 {\vartheta} > {\chi} > 0, \ {\zeta} > \delta > 0 , \Phi_{{\kappa}}\in C(\mbox{ \mathbb{R} }^{+}, \mbox{ \mathbb{R} }^{+}) , E, ϱ1, ϱ2≥0 \mathcal {E}, \ \varrho_{1}, \ \varrho_{2} \geq 0 a.e. on Q=[0,1] Q = [0, 1] , E, ϱ1, ϱ2∈L1(0,1) \mathcal {E}, \ \varrho_{1}, \ \varrho_{2} \in L^{1}(0, 1) and \digamma:[0, 1]\times \mbox{ \mathbb{R} }^{+}\rightarrow \mbox{ \mathbb{R} }^{+} , \mbox{ \mathbb{R} }^{+} = [0, +\infty) . By using Krasnosel skii's fixed point theorem for discontinuous operators on cones, some sufficient conditions for the existence of single or multiple positive solutions for the above discontinuous differential system are established. An example is given to confirm the main results in the end.
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... Nowadays impulsive differential equations are attracting a lot of attention. They appear in the study of several real world problems (see, for instance, [1,2,15]). In general, it is well known that several natural phenomena are driven by differential equations, but the description of some real world problems subjected to sudden changes in their stated became very interesting from the mathematical point of view because they should be described considering systems of differential equations with impulses. ...
RETRACTED ARTICLE: Second-order impulsive differential systems of mixed type: oscillation theorems
Article
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  • Sep 2022
  • BOUND VALUE PROBL
In this paper, we obtain necessary and sufficient conditions for the oscillation of solutions to a second-order neutral differential equation with impulses. Two examples are provided to show the effectiveness and feasibility of the main results. Our main tool is Lebesgue’s dominated convergence theorem.
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... IDEs are related to mechanical systems,theoretical physics, chemistry, control theory and so on. Hence, they are regarded as an effective mathematical tool to solve some real-world problems in the applied sciences (see, for instance, [1][2][3][4]). Many authors are devoted to the existence of solutions to IDEs. ...
Multiple Solutions for a Class of BVPs of Second-Order Discontinuous Differential Equations with Impulse Effects
Article
Full-text available
  • Jul 2022
This paper deals with a class of boundary value problems of second-order differential equations with impulses and discontinuity. The existence of single or multiple positive solutions to discontinuous differential equations with impulse effects is established by using the nonlinear alternative of Krasnoselskii’s fixed point theorem for discontinuous operators on cones. Finally, an example is given to illustrate the main results.
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... where x = x(t) and y = y(t) represent the densities of two species at time t, respectively; a 1 and a 2 represent the intrinsic growth rate of the two species, b 1 and b 2 are positive constants, p ≥ 1, c 1 > 0. The authors investigated the properties of equilibria of the system. It is well known that delays can have great impact on the dynamics of a system, for example, delays can induce oscillations [Agarwal et al., 2014], induce chaotic behaviors [Kuang, 1993], delays can cause the loss of stability of equilibria and induce bifurcation phenomena [Niu & Jiang, 2013;Yuan et al., 2015;Su & Zou, 2014;Wang & Wei, 2019;Chen & Wei, 2015;Zhang et al., 2009;Yan & Chu, 2006;Liao et al., 2014;Zhang et al., 2010]. In 2010, Zhang et al. [2010] proposed the following predatorprey model with a discrete delay and a distributed delay ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dx(t) dt = x(t)((r 1 − a 11 x(t) − a 12 y(t − τ )), ...
Bifurcation Analysis for Two-Species Commensalism (Amensalism) Systems with Distributed Delays
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... Nowadays, the study of qualitative properties of ordinary differential equations attracts considerable attention from the scientific community due to numerous applications of them to several contexts, such as Biology, Physics, Chemistry, and Dynamical Systems. For some details related to the recent studies on oscillation and non-oscillation properties, exponential stability, instability, existence of unbounded solutions of the equations under consideration, we refer the reader to the books [1,2]. ...
Some conditions for the oscillation of second-order differential equations with several mixed delays
Article
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  • Feb 2022
  • J FIX POINT THEORY A
In this work, we obtain necessary and sufficient conditions for the oscillation of the solutions to a second-order neutral differential equation with mixed delays. Two examples are provided to show effectiveness and feasibility of main results. Our main tool is the Lebesgue’s Dominated Convergence theorem.
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... Then, models describing viscoelastic bodies colliding systems with delay and impulses are more appropriate (see [1] and references therein for a review). The models appear in the study of several real-world problems (see, for instance, [2][3][4]). In general, it is well-known that several natural phenomena are driven by impulsive differential equations. ...
On Nonlinear Forced Impulsive Differential Equations under Canonical and Non-Canonical Conditions
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The asymptotic behavior and oscillation for a class of third-order nonlinear delay dynamic equations
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  • ACTA MATH SCI
In this paper, we consider a class of third-order nonlinear delay dynamic equations. First, we establish a Kiguradze-type lemma and some useful estimates. Second, we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero. Third, we obtain new oscillation criteria by employing the Pötzsche chain rule. Then, using the generalized Riccati transformation technique and averaging method, we establish the Philos-type oscillation criteria. Surprisingly, the integral value of the Philos-type oscillation criteria, which guarantees that all unbounded solutions oscillate, is greater than θ4(t1, T). The results of Theorem 3.5 and Remark 3.6 are novel. Finally, we offer four examples to illustrate our results.
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Second‐order impulsive differential systems with mixed delays: Oscillation theorems
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  • MATH METHOD APPL SCI
In this work, we obtain necessary and sufficient conditions for the oscillation of solutions of a second‐order neutral differential equation with impulses. Two examples are provided to show the effectiveness and feasibility of the main results. Our main tool is the Lebesgue's Dominated Convergence theorem.
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Oscillation Theory of Dynamic Equations on Time Scales: Second and Third Orders
Book
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  • Apr 2010
The study of dynamic equations on time scales, which goes back to its founder Stefan Hilger is an area of mathematics and has been created in order to unify the study of differential and difference equations. The oscillation theory as a part of the qualitative theory of dynamic equations on time scales has been developed rapidly in the past ten years. The extensive application prospect facilitates the development of this field. In fact there are some applications of dynamic equations in population dynamics, quantum
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Global stability and chaos in a population model with piecewise constant arguments
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  • APPL MATH COMPUT
Sufficient conditions are obtained for the global stability of the positive equilibrium of dx/dt=rx(t){1−ax(t)−b∑j=0∞cjx([t−j])}. It is shown that for certain special cases, solutions of the equation can have chaotic behaviour through period doubling bifurcations.
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Oscillation and global attractivity in a nonlinear delay periodic model of Respiratory Dynamics
Article
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In this paper, we shall consider the nonlinear delay differential equation where m and n are positive integers, and V(t) and λ(t) are positive periodic functions of period ω. In the nondelay case, we shall show that (∗) has a unique positive periodic solution (t) and provides sufficient conditions for the global attractivity of (t). In the delay case, we shall present sufficient conditions for the oscillation of all positive solutions of (∗) about (t) and establish sufficient conditions for the global attractivity of (t).
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Global attractivity for a differential-difference population model
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On Oscillation of Equations with Distributed Delay
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For the scalar delay differential equation with a distributed delay \dot{x} (t) + \int ^t_{– \infty} x(s)d_sR(t,s) = f(t) (t > t_o) a connection between the propertiesnon-oscillationpositiveness of the fundamental functionexistence of a non-negative solution for a certain nonlinear integral inequalityis established. This enables to obtain comparison theorems and explicit non-oscillation and oscillation conditions being generalizations of some known results for delay equations and integro-differential equations and leads to oscillation results for equations with infinite number of delays.
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