The entry-exit theorem for the phenomenon of delay of stability loss for certain types of slow-fa... more The entry-exit theorem for the phenomenon of delay of stability loss for certain types of slow-fast planar systems plays a key role in establishing existence of limit cycles that exhibit relaxation oscillations. The general existing proofs of this theorem depend on Fenichel's geometric singular perturbation theory and blow-up techniques. In this work, we give a short and elementary proof of the entry-exit theorem based on a direct study of asymptotic formulas of the underlying solutions. We employ this theorem to a broad class of slow-fast planar systems to obtain existence, global uniqueness and asymptotic orbital stability of relaxation oscillations. The results are then applied to a diffusive predator-prey model with Holling type II functional response to establish periodic traveling wave solutions. Furthermore, we extend our work to another class of slow-fast systems that can have multiple orbits exhibiting relaxation oscillations, and subsequently apply the results to a two timescale Holling-Tanner predator-prey model with Holling type IV functional response. It is generally assumed in the literature that the non-trivial equilibrium points exist uniquely in the interior of the domains bounded by the relaxation oscillations; we do not make this assumption in this paper.
Discrete and Continuous Dynamical Systems - B, 2021
A two-dimensional stage-structured model for the interactive wild and sterile mosquitoes is deriv... more A two-dimensional stage-structured model for the interactive wild and sterile mosquitoes is derived where the wild mosquito population is composed of larvae and adult classes and only sexually active sterile mosquitoes are included as a function given in advance. The strategy of constant releases of sterile mosquitoes is considered but periodic and impulsive releases are more focused on. Local stability of the origin and the existence of a positive periodic solution are investigated. While mathematical analysis is more challenging, numerical examples demonstrate that the model dynamics, determined by thresholds of the release amount and the release waiting period, essentially match the dynamics of the alike one-dimensional models. It is also shown that richer dynamics are exhibited from the two-dimensional stage-structured model.
In this article we are interested in the existence of positive classical solutions of −∆u + a(x) ... more In this article we are interested in the existence of positive classical solutions of −∆u + a(x) • ∇u + V (x)u = u p + γu q in Ω u = 0 on ∂Ω, (1) and −∆u + a(x) • ∇u + V (x)u = u p + γ|∇u| q in Ω u = 0 on ∂Ω, (2) where Ω is a smooth exterior domain in R N in the case of N ≥ 4, p > N +1 N −3 and γ ∈ R. We assume that V is a smooth nonnegative potential and a(x) is a smooth vector field, both of which satisfy natural decay assumptions. Under suitable assumptions on q we prove the existence of an infinite number of positive classical solutions. We also consider the case of N +2 N −2 < p < N +1 N −3 under further symmetry assumptions on Ω, a and V .
To investigate the impact of periodic and impulsive releases of sterile mosquitoes on the interac... more To investigate the impact of periodic and impulsive releases of sterile mosquitoes on the interactive dynamics between wild and sterile mosquitoes, we adapt the new idea where only those sexually active sterile mosquitoes are included in the modelling process and formulate new models with time delay. We consider different release strategies and compare their model dynamics. Under certain conditions, we derive corresponding model formulations and prove the existence of periodic solutions for some of those models. We provide numerical examples to demonstrate the dynamical complexity of the models and propose further studies.
To prevent the transmissions of mosquito-borne diseases (e.g., malaria, dengue fever), recent wor... more To prevent the transmissions of mosquito-borne diseases (e.g., malaria, dengue fever), recent works have considered the problem of using the sterile insect technique to reduce or eradicate the wild mosquito population. It is important to consider how reproductive advantage of the wild mosquito population offsets the success of population replacement. In this work, we explore the interactive dynamics of the wild and sterile mosquitoes by incorporating the delay in terms of the growth stage of the wild mosquitoes. We analyze (both analytically and numerically) the role of time delay in two different ways of releasing sterile mosquitoes. Our results demonstrate that in the case of constant release rate, the delay does not affect the dynamics of the system and every solution of the system approaches to an equilibrium point; while in the case of the release rate proportional to the wild mosquito populations, the delay has a large effect on the dynamics of the system, namely, for some parameter ranges, when the delay is small, every solution of the system approaches to an equilibrium point; but as the delay increases, the solutions of the system exhibit oscillatory behavior via Hopf bifurcations. Numerical examples and bifurcation diagrams are also given to demonstrate rich dynamical features of the model in the latter release case.
In this short note we establish global stability results for a four- dimensional nonlinear system... more In this short note we establish global stability results for a four- dimensional nonlinear system that was developed in modeling a tick-borne disease by H.D. Gafi and L.J. Gross (Bull. Math. Biol., 69 (2007), 265{288) where local stability results were obtained. These results provide the parameter ranges for controlling long-term population and disease dynamics. 1. Introduction. In the United States,
We discuss the existence and the uniqueness of travelling wave solutions for a tissue interaction... more We discuss the existence and the uniqueness of travelling wave solutions for a tissue interaction model on skin pattern formation proposed by Cruywagen and Murray. The geometric theory of singular perturbations is employed.
A shooting approach to layers and chaos for a forced Duffing equation
We study equilibrium solutions for the problem u t=ɛ 2u xx-u 3+λu+cos x, u x(0, t)=u x(1, t)=0. U... more We study equilibrium solutions for the problem u t=ɛ 2u xx-u 3+λu+cos x, u x(0, t)=u x(1, t)=0. Using a shooting method we find solutions for all nonzero ɛ. For small ɛ we add to the solutions found by previous authors, especially Angenent, Mallet-Paret and Peletier, and Hale and Sakamoto, and also give new elementary ode proofs of their results. Among the new results is the existence of internal layer-type solutions. Considering the ode satisfied by equilibria, but on an infinite interval, we obtain chaos results for λ⩾λ 0= {3}/{2 2/3} and 0<ɛ⩽ {1}/{4}. We also consider the problem of bifurcation of solutions as λ increases.
Transactions of the American Mathematical Society, 2006
and ε > 0 is sufficiently small, on an interval [0, L] with boundary conditionsu = 0 at x = 0, L.... more and ε > 0 is sufficiently small, on an interval [0, L] with boundary conditionsu = 0 at x = 0, L. All solutions with an ε independent number of oscillations are analyzed. Existence of complicated patterns of layers and spikes is proved, and their Morse index is determined. It is observed that the results extend to
By constructing an invariant set in the three dimensional space, we establish the existence of tr... more By constructing an invariant set in the three dimensional space, we establish the existence of traveling wave solutions to a reaction-diffusionchemotaxis model describing biological processes such as the bacterial chemotactic movement in response to oxygen and the initiation of angiogenesis. The minimal wave speed is shown to exist and the role of each process of reaction, diffusion and chemotaxis in the wave propagation is investigated. Our results reveal three essential biological implications: (1) the cell growth increases the wave speed; (2) the chemotaxis must be strong enough to make a contribution to the increment of the wave speed; (3) the diffusion rate plays a role in increasing the wave speed only when the cell growth is present.
and ε > 0 is sufficiently small, on an interval [0, L] with boundary conditionsu = 0 at x = 0, L.... more and ε > 0 is sufficiently small, on an interval [0, L] with boundary conditionsu = 0 at x = 0, L. All solutions with an ε independent number of oscillations are analyzed. Existence of complicated patterns of layers and spikes is proved, and their Morse index is determined. It is observed that the results extend to
Reaction, diffusion and chemotaxis in wave propagation
ABSTRACT By constructing an invariant set in the three dimensional space, we establish the existe... more ABSTRACT By constructing an invariant set in the three dimensional space, we establish the existence of traveling wave solutions to a reaction-diffusion- chemotaxis model describing biological processes such as the bacterial chemo- tactic movement in response to oxygen and the initiation of angiogenesis. The minimal wave speed is shown to exist and the role of each process of reaction, diffusion and chemotaxis in the wave propagation is investigated. Our results reveal three essential biological implications: (1) the cell growth increases the wave speed; (2) the chemotaxis must be strong enough to make a contribu- tion to the increment of the wave speed; (3) the diffusion rate plays a role in increasing the wave speed only when the cell growth is present.
Mosquito-Stage-Structured Malaria Models and Their Global Dynamics
SIAM Journal on Applied Mathematics, 2012
ABSTRACT We formulate mosquito-stage-structured, continuous-time, compartmental, malaria models w... more ABSTRACT We formulate mosquito-stage-structured, continuous-time, compartmental, malaria models which include four distinct metamorphic stages of mosquitoes. We derive a formula for the reproductive number of infection and investigate the existence of endemic equilibria. We determine conditions under which the models undergo either forward or backward bifurcation. We carry out rigorous mathematical analysis on the model dynamics by proving the global stability of the infectionfree equilibrium while a forward bifurcation occurs, and the global stability of the endemic equilibrium as the reproductive number is greater than one in certain cases. Our study can be also applied to other diseases such as Chikungunya fever.
We study a bioremediation model that arises in restoring ground water and soil contaminated with ... more We study a bioremediation model that arises in restoring ground water and soil contaminated with organic pollutants. It describes an in situ bioredimedation scenario in which a sorbing substrate of contaminated soil is degraded by indigenous microorganisms in the presence of an injected nonsorbing electron acceptor. The model relates to the coupling of the advection, dispersion, and biological reaction simultaneously for the substrate, electron acceptor, and the total biomass by two advection-reaction-diffusion equations and an ODE.
The entry-exit theorem for the phenomenon of delay of stability loss for certain types of slow-fa... more The entry-exit theorem for the phenomenon of delay of stability loss for certain types of slow-fast planar systems plays a key role in establishing existence of limit cycles that exhibit relaxation oscillations. The general existing proofs of this theorem depend on Fenichel's geometric singular perturbation theory and blow-up techniques. In this work, we give a short and elementary proof of the entry-exit theorem based on a direct study of asymptotic formulas of the underlying solutions. We employ this theorem to a broad class of slow-fast planar systems to obtain existence, global uniqueness and asymptotic orbital stability of relaxation oscillations. The results are then applied to a diffusive predator-prey model with Holling type II functional response to establish periodic traveling wave solutions. Furthermore, we extend our work to another class of slow-fast systems that can have multiple orbits exhibiting relaxation oscillations, and subsequently apply the results to a two timescale Holling-Tanner predator-prey model with Holling type IV functional response. It is generally assumed in the literature that the non-trivial equilibrium points exist uniquely in the interior of the domains bounded by the relaxation oscillations; we do not make this assumption in this paper.
Discrete and Continuous Dynamical Systems - B, 2021
A two-dimensional stage-structured model for the interactive wild and sterile mosquitoes is deriv... more A two-dimensional stage-structured model for the interactive wild and sterile mosquitoes is derived where the wild mosquito population is composed of larvae and adult classes and only sexually active sterile mosquitoes are included as a function given in advance. The strategy of constant releases of sterile mosquitoes is considered but periodic and impulsive releases are more focused on. Local stability of the origin and the existence of a positive periodic solution are investigated. While mathematical analysis is more challenging, numerical examples demonstrate that the model dynamics, determined by thresholds of the release amount and the release waiting period, essentially match the dynamics of the alike one-dimensional models. It is also shown that richer dynamics are exhibited from the two-dimensional stage-structured model.
In this article we are interested in the existence of positive classical solutions of −∆u + a(x) ... more In this article we are interested in the existence of positive classical solutions of −∆u + a(x) • ∇u + V (x)u = u p + γu q in Ω u = 0 on ∂Ω, (1) and −∆u + a(x) • ∇u + V (x)u = u p + γ|∇u| q in Ω u = 0 on ∂Ω, (2) where Ω is a smooth exterior domain in R N in the case of N ≥ 4, p > N +1 N −3 and γ ∈ R. We assume that V is a smooth nonnegative potential and a(x) is a smooth vector field, both of which satisfy natural decay assumptions. Under suitable assumptions on q we prove the existence of an infinite number of positive classical solutions. We also consider the case of N +2 N −2 < p < N +1 N −3 under further symmetry assumptions on Ω, a and V .
To investigate the impact of periodic and impulsive releases of sterile mosquitoes on the interac... more To investigate the impact of periodic and impulsive releases of sterile mosquitoes on the interactive dynamics between wild and sterile mosquitoes, we adapt the new idea where only those sexually active sterile mosquitoes are included in the modelling process and formulate new models with time delay. We consider different release strategies and compare their model dynamics. Under certain conditions, we derive corresponding model formulations and prove the existence of periodic solutions for some of those models. We provide numerical examples to demonstrate the dynamical complexity of the models and propose further studies.
To prevent the transmissions of mosquito-borne diseases (e.g., malaria, dengue fever), recent wor... more To prevent the transmissions of mosquito-borne diseases (e.g., malaria, dengue fever), recent works have considered the problem of using the sterile insect technique to reduce or eradicate the wild mosquito population. It is important to consider how reproductive advantage of the wild mosquito population offsets the success of population replacement. In this work, we explore the interactive dynamics of the wild and sterile mosquitoes by incorporating the delay in terms of the growth stage of the wild mosquitoes. We analyze (both analytically and numerically) the role of time delay in two different ways of releasing sterile mosquitoes. Our results demonstrate that in the case of constant release rate, the delay does not affect the dynamics of the system and every solution of the system approaches to an equilibrium point; while in the case of the release rate proportional to the wild mosquito populations, the delay has a large effect on the dynamics of the system, namely, for some parameter ranges, when the delay is small, every solution of the system approaches to an equilibrium point; but as the delay increases, the solutions of the system exhibit oscillatory behavior via Hopf bifurcations. Numerical examples and bifurcation diagrams are also given to demonstrate rich dynamical features of the model in the latter release case.
In this short note we establish global stability results for a four- dimensional nonlinear system... more In this short note we establish global stability results for a four- dimensional nonlinear system that was developed in modeling a tick-borne disease by H.D. Gafi and L.J. Gross (Bull. Math. Biol., 69 (2007), 265{288) where local stability results were obtained. These results provide the parameter ranges for controlling long-term population and disease dynamics. 1. Introduction. In the United States,
We discuss the existence and the uniqueness of travelling wave solutions for a tissue interaction... more We discuss the existence and the uniqueness of travelling wave solutions for a tissue interaction model on skin pattern formation proposed by Cruywagen and Murray. The geometric theory of singular perturbations is employed.
A shooting approach to layers and chaos for a forced Duffing equation
We study equilibrium solutions for the problem u t=ɛ 2u xx-u 3+λu+cos x, u x(0, t)=u x(1, t)=0. U... more We study equilibrium solutions for the problem u t=ɛ 2u xx-u 3+λu+cos x, u x(0, t)=u x(1, t)=0. Using a shooting method we find solutions for all nonzero ɛ. For small ɛ we add to the solutions found by previous authors, especially Angenent, Mallet-Paret and Peletier, and Hale and Sakamoto, and also give new elementary ode proofs of their results. Among the new results is the existence of internal layer-type solutions. Considering the ode satisfied by equilibria, but on an infinite interval, we obtain chaos results for λ⩾λ 0= {3}/{2 2/3} and 0<ɛ⩽ {1}/{4}. We also consider the problem of bifurcation of solutions as λ increases.
Transactions of the American Mathematical Society, 2006
and ε > 0 is sufficiently small, on an interval [0, L] with boundary conditionsu = 0 at x = 0, L.... more and ε > 0 is sufficiently small, on an interval [0, L] with boundary conditionsu = 0 at x = 0, L. All solutions with an ε independent number of oscillations are analyzed. Existence of complicated patterns of layers and spikes is proved, and their Morse index is determined. It is observed that the results extend to
By constructing an invariant set in the three dimensional space, we establish the existence of tr... more By constructing an invariant set in the three dimensional space, we establish the existence of traveling wave solutions to a reaction-diffusionchemotaxis model describing biological processes such as the bacterial chemotactic movement in response to oxygen and the initiation of angiogenesis. The minimal wave speed is shown to exist and the role of each process of reaction, diffusion and chemotaxis in the wave propagation is investigated. Our results reveal three essential biological implications: (1) the cell growth increases the wave speed; (2) the chemotaxis must be strong enough to make a contribution to the increment of the wave speed; (3) the diffusion rate plays a role in increasing the wave speed only when the cell growth is present.
and ε > 0 is sufficiently small, on an interval [0, L] with boundary conditionsu = 0 at x = 0, L.... more and ε > 0 is sufficiently small, on an interval [0, L] with boundary conditionsu = 0 at x = 0, L. All solutions with an ε independent number of oscillations are analyzed. Existence of complicated patterns of layers and spikes is proved, and their Morse index is determined. It is observed that the results extend to
Reaction, diffusion and chemotaxis in wave propagation
ABSTRACT By constructing an invariant set in the three dimensional space, we establish the existe... more ABSTRACT By constructing an invariant set in the three dimensional space, we establish the existence of traveling wave solutions to a reaction-diffusion- chemotaxis model describing biological processes such as the bacterial chemo- tactic movement in response to oxygen and the initiation of angiogenesis. The minimal wave speed is shown to exist and the role of each process of reaction, diffusion and chemotaxis in the wave propagation is investigated. Our results reveal three essential biological implications: (1) the cell growth increases the wave speed; (2) the chemotaxis must be strong enough to make a contribu- tion to the increment of the wave speed; (3) the diffusion rate plays a role in increasing the wave speed only when the cell growth is present.
Mosquito-Stage-Structured Malaria Models and Their Global Dynamics
SIAM Journal on Applied Mathematics, 2012
ABSTRACT We formulate mosquito-stage-structured, continuous-time, compartmental, malaria models w... more ABSTRACT We formulate mosquito-stage-structured, continuous-time, compartmental, malaria models which include four distinct metamorphic stages of mosquitoes. We derive a formula for the reproductive number of infection and investigate the existence of endemic equilibria. We determine conditions under which the models undergo either forward or backward bifurcation. We carry out rigorous mathematical analysis on the model dynamics by proving the global stability of the infectionfree equilibrium while a forward bifurcation occurs, and the global stability of the endemic equilibrium as the reproductive number is greater than one in certain cases. Our study can be also applied to other diseases such as Chikungunya fever.
We study a bioremediation model that arises in restoring ground water and soil contaminated with ... more We study a bioremediation model that arises in restoring ground water and soil contaminated with organic pollutants. It describes an in situ bioredimedation scenario in which a sorbing substrate of contaminated soil is degraded by indigenous microorganisms in the presence of an injected nonsorbing electron acceptor. The model relates to the coupling of the advection, dispersion, and biological reaction simultaneously for the substrate, electron acceptor, and the total biomass by two advection-reaction-diffusion equations and an ODE.
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