Effects of Refuge Prey on Stability of the Prey-Predator Model Subject to Immigrants: A Mathematical Modelling Approach

Prey-predator system is enormously complex and nonlinear interaction between species. Such complexity regularly requires development of new approaches which involves more factors in analysis of its population dynamics. In this paper, we formulate a modified Lotka-Volterra model that incorporates factors such as refuge prey and immigrants. We investigate the effects of refuge prey and immigrants by varying the refuge factor, with and without immigrants. The results show that with Holling’s type I functional response, the proposed model is asymptotically convergent when a refuge prey factor is introduced. Moreover, with Holling’s type II functional response, the proposed mathematical model is unstable and does not converge. However, with Holling’s type III functional response in a system, the proposed mathematical model is asymptotically stable. These results point out the following remarks: The effects of refuge prey on stability of the dynamical system vary depending on the type of functional response, and when the predator population increases, the likelihood of prey extinction declines when the proportion of preys in refuge population increases. Hence, the factor of refuge prey is crucial for controlling the population of the predator and obtaining balances between prey and predator in the ecosystem. Keywords: Refuge prey, stability, prey-predator, immigrants, Mathematical modellin

Optimal Harvesting for an Age-Spatial-Structured Population Dynamic Model with External Mortality

We study an optimal harvesting for a nonlinear age-spatial-structured population dynamic model, where the dynamic system contains an external mortality rate depending on the total population size. The total mortality consists of two types: the natural, and external mortality and the external mortality reflects the effects of external environmental causes. We prove the existence and uniqueness of solutions for the population dynamic model. We also derive a sufficient condition for optimal harvesting and some necessary conditions for optimality in an optimal control problem relating to the population dynamic model. The results may be applied to an optimal harvesting for some realistic biological models

Dynamic Analysis of a Two-Language Competitive Model with Control Strategies

The dynamic behavior of a two-language competitive model is analyzed systemically in this paper. By the linearization and the Bendixson-Dulac theorem on dynamical system, some sufficient conditions on the globally asymptotical stability of the trivial equilibria and the existence and the stability of the positive equilibrium of this model are presented. Nextly, in order to protect the endangered language, an optimal control problem relative to this model is explored. We derive some necessary conditions to solve the optimal control problem and present some numerical simulations using a Runge-Kutta fourth-order method. Finally, the languages competitive model is extended to this model assessing the impact of state-dependent pulse control strategy. Using the Poincaré map, differential inequality, and method of qualitative analysis, we prove the existence and stability of positive order-1 periodic solution for this control model. Numerical simulations are carried out to illustrate the main results and the feasibility of state-dependent impulsive control strategy.This paper has been partially supported by the Scientific Research Programmes of Colleges in Xinjiang (Grant no. XJEDU2011S08), the Natural Science Foundation of Xinjiang (Grant no. 2011211B08), the National Natural Science Foundation of China (Grants nos. 11001235, 11271312, and 11261056), the China Postdoctoral Science Foundation (Grants nos. 20110491750 and 2012T50836), and the Ministerio de Economia y Competitividad (Spain), Project no. MTM2010-15314 and has been co-financed by the European Community Fund FEDERS

Optimal Harvesting for an Age-Spatial-Structured Population Dynamic Model with External Mortality

We study an optimal harvesting for a nonlinear age-spatial-structured population dynamic model, where the dynamic system contains an external mortality rate depending on the total population size. The total mortality consists of two types: the natural, and external mortality and the external mortality reflects the effects of external environmental causes. We prove the existence and uniqueness of solutions for the population dynamic model. We also derive a sufficient condition for optimal harvesting and some necessary conditions for optimality in an optimal control problem relating to the population dynamic model. The results may be applied to an optimal harvesting for some realistic biological models

Stability Analysis of a Vector-Borne Disease with Variable Human Population

A mathematical model of a vector-borne disease involving variable human population is analyzed. The varying population size includes a term for disease-related deaths. Equilibria and stability are determined for the system of ordinary differential equations. If R0≤1, the disease-“free” equilibrium is globally asymptotically stable and the disease always dies out. If R0>1, a unique “endemic” equilibrium is globally asymptotically stable in the interior of feasible region and the disease persists at the “endemic” level. Our theoretical results are sustained by numerical simulations

The Association Between Masticatory Function Assessment and Masseter Muscle Thickness in the Elderly

Purpose: Purpose: This study investigated the association between the objective indicator of masticatory function assessment and the masseter muscle thickness (MMT) using ultrasound imaging. Methods: Methods: A total of 99 subjects (males: 24, females: 75, mean age: 76) were analyzed. The maximum bite force (MBF) was measured with a pressure-sensitive sheet and an image scanner. The mixing ability index (MAI) was calculated by image analysis after asking the subjects to chew a wax specimen. The MMT during rest and clenching were obtained with a diagnostic ultrasound system, and the difference in MMT during rest and MMT during clenching was defined as the difference in masseter muscle thickness (DMMT). Multiple regression analysis was performed to determine the independent variables affecting MBF and MAI. Results: Results: The MBF showed correlation with the number of remaining teeth (β=0.346, p=0.002) and DMMT (β=0.251, p=0.011). The MAI correlated with only the number of remaining teeth (β=0.476, p<0.001). Conclusions: Conclusions: The DMMT reflects the state of masseter muscle contraction, and can be used as a predictor as well as the number of teeth when assessing masticatory function.22Nkc

Mathematical Analysis of a Malaria Model with Partial Immunity to Reinfection

A deterministic model with variable human population for the transmission dynamics of malaria disease, which allows transmission by the recovered humans, is first developed and rigorously analyzed. The model reveals the presence of the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with one or more stable endemic equilibria when the associated reproduction number is less than unity. This phenomenon may arise due to the reinfection of host individuals who recovered from the disease. The model in an asymptotical constant population is also investigated. This results in a model with mass action incidence. A complete global analysis of the model with mass action incidence is given, which reveals that the global dynamics of malaria disease with reinfection is completely determined by the associated reproduction number. Moreover, it is shown that the phenomenon of backward bifurcation can be removed by replacing the standard incidence function with a mass action incidence. Graphical representations are provided to study the effect of reinfection rate and to qualitatively support the analytical results on the transmission dynamics of malaria

Global Existence and Energy Decay Rates for a Kirchhoff-Type Wave Equation with Nonlinear Dissipation

The first objective of this paper is to prove the existence and uniqueness of global solutions for a Kirchhoff-type wave equation with nonlinear dissipation of the form Kuⁿ + (|¹/²|²) + (')=0 under suitable assumptions on , , (⋅), and (⋅). Next, we under some growth conditions on the nonlinear dissipation . Lastly, numerical simulations in order to verify the analytical results are given

Epileptic nystagmus: A case report and systematic review

AbstractPurposeWe aimed to define the characteristics of epileptic nystagmus and correlate those with other clinical findings in a large number of patients.MethodsWe report a patient with epileptic nystagmus and additionally reviewed the reported clinical features of 36 more patients through a systematic literature search. We analyzed the characteristics of epileptic nystagmus and attempted correlations of those with alertness of the patients and epileptic foci on EEG.ResultsAll 33 patients with unilateral horizontal nystagmus showed nystagmus beating away from the side of ictal discharges. Epileptic nystagmus was preceded by gaze deviation in 21 patients, with contraversive in 19 and ipsiversive in 2. Seizures associated with epileptic nystagmus were mostly focal (25/29, 86.2%) with or without loss of awareness. Ictal discharges originated from the occipital (n=16), parietal (n=9), temporo-occipital (n=6), frontal (n=4), and temporal (n=3) areas, and two patients had multiple epileptic foci. Seizures were usually symptomatic (24/37, 64.9%). The presence of preceding gaze deviation and midline crossing of the nystagmus did not correlate with the ictal onset zone or alertness of the patients. Recording of epileptic nystagmus was available only in 6 patients, and the epileptic nystagmus could be localized to the saccadic areas in two and to the smooth pursuit areas in another two. Two patients showed the features of epileptic nystagmus from both areas.ConclusionEven though the localizing value of epileptic nystagmus seems limited in previous reports, the fast phase of epileptic nystagmus was almost always directed away from the epileptic focus that mostly arose from the posterior part of the cerebral hemisphere