Applied Dynamical Systems Seminar (Winter 2017)

Organizer: Dr. Yuan Yuan

Time and Location: 1:00pm to 2:00pm, Tuesday, HH-3017

Speakers and Abstracts

• 1. Jan. 24, Xiunan Wang (MUN)     ``Malaria Transmission Model with Temperature-dependent Incubation Period''

Malaria is an infectious disease caused by Plasmodium parasites and is transmitted among humans by female Anopheles mosquitoes. Climate factors have significant impact on both mosquito life cycle and parasite development. To consider the temperature sensitivity of the extrinsic incubation period (EIP) of malaria parasites, we formulate a delay differential equations model with a periodic time delay. We derive the basic reproduction ratio R0 and establish a threshold type result on the global dynamics in terms of R0, that is, the unique disease-free periodic solution is globally attractive if R0<1; and the model system admits a unique positive periodic solution which is globally attractive if R0>1. Numerically, we parameterize the model with data from Maputo Province, Mozambique and simulate the long term behavior of solutions. The simulation result is consistent with the obtained analytic result. In addition, we find that using the time-averaged EIP may result in underestimation of the basic reproduction ratio. This talk is based on a joint work with Prof. Xiao-Qiang Zhao.



• 2. Feb. 7, Peiluan Li (Henan University of Science and Technology,China)     ``The boundary value problems for a coupled system of fractional differential equations''

Using the variational methods, we investigate the solutions to the boundary value problems for a coupled system of fractional order differential equations. First, we obtain the existence of at least one weak solution by the minimization result due to Mawhin and Willem. Then, the existence criteria of infinitely many solutions are established by a critical point theorem due to Rabinowitz. At last, some examples are also provided to illustrate the results.



• 3. Feb. 21, Lei Zhang (MUN/University of Science and Technology of China)     ``The Principal Eigenvalue for Degenerate Periodic Reaction-Diffusion Systems''

In this talk, I will report our recent research on the theory of the principal eigenvalue for an eigenvalue problem associated with a linear time-periodic parabolic cooperative system with some zero diffusion coefficients. We use a generalized Krein-Rutman theorem to overcome the main difficulty induced by the lack of compactness for the solution maps. We first review the Krein-Rutman theorem and Nussbaum's generalization and then present the main results. The developed theory is also applied to a benthic-drift model for a stream population to obtain a threshold type result on its global dynamics in terms of the basic reproduction ratio. This talk is based on a joint work with Drs. Xing Liang and Xiaoqiang Zhao.



• 4. Feb. 28, Zhenguo Bai (Xidian University, Xi'an, China)     ``A reaction-diffusion malaria model with seasonality and incubation period''

In this talk, I will consider a time-periodic reaction-diffusion model which incorporates seasonality, spatial heterogeneity and the extrinsic incubation period (EIP) of the parasite. The basic reproduction number $R_0$ is derived, and it is shown that the disease-free periodic solution is globally attractive if $R_0<1$, while there is an endemic periodic solution and the disease is uniformly persistent if $R_0>1$. Numerical simulations indicate that prolonging the EIP and increasing population mobility may be helpful in the disease control, while spatial heterogeneity of the disease transmission coefficient may increase the disease burden. This talk is based on a joint work with Drs. Rui Peng and Xiao-qiang Zhao.



• 5. Mar. 10, (Joint with the Departmental Colloquium) Dr. Frithjof Lutscher (University of Ottawa)     ``Behavioural responses to resource heterogeneity can accelerate biological invasions''

After an invasive species is introduced, the abundance and spatial distribution of resources in a landscape and the behavioural response of individuals determine whether and how fast it spreads in the given environment. It is therefore of interest whether and how landscape manipulations can be used to slow invasive species is of great interest. Various ideas in this direction are being discussed as management options in forest ecosystems, for example tree removal, thinning, and increasing tree diversity. Recent experiments show individual-level behavioural movement changes in response to a spatially heterogeneous resource distribution. This behaviour needs to be included into management considerations to correctly predict the effects of any control measures.

We derive a novel model for insect-host dynamics that includes three common behavioural aspects of foraging: higher movement rate in resource-poor areas, lower ovipositioning rate in resource-poor areas, and movement preference for resource-rich areas. We derive appropriate dispersal kernels from our movement model and use them to project the insect population density from one year to the next. Using a mix of analysis and simulation, we explore how several management options affect the ability and the speed of the invasive species. We parameterize our model and illustrate our results with data for Emerald ash borer, a recent highly destructive forest pest in North America.

We show that each of the three basic movement behaviours can increase the speed of invasion in a source-sink landscape above that in a homogeneous landscape with larger overall resource availability. Our results highlight the importance of empirical work on movement behaviour in different landscape types and near the interface between types.



• 6. Mar. 28, Isam Al-Darabsah (MUN)     ``A Periodic Disease Transmission Model with Asymptomatic Carriage and Latency Periods. ''

In this talk, we propose a periodic disease transmission model with two delays in incubation and asymptomatic carriage periods. We will identify the basic reproduction ratio R_0 for the model; obtain the global attractivity of the disease-free state when R_0<1 and discuss the disease persistence when R_0>1. We will also explore the coexistence and uniqueness of endemic state in the system with constants coefficients. Numerically, we provide a numerical algorithm to calculate R_0; present a case study regarding the meningococcal meningitis disease transmission and discuss the influence of carriers on R_0. This is a joint work with Dr. Yuan Yuan.


• 7. Apr. 18, (Joint with the Departmental Colloquium) Peiluan Li (Henan University of Science and Technology,China)     `` Solutions for Impulsive Fractional Differential Equations via Variational Methods ''

In this talk, we investigate the boundary value problems of impulsive fractional order differential equations. After obtaining the existence of at least one solution from the minimization result, we establish the existence results of at least triple solutions by the variational methods and a very recent critical point theorem due to Bonanno and Marano. In addition we give the existence criteria of infinitely many solutions based onthe variational methods and a critical point theorem. Some examples are provided to demonstrate the application of the main results.