Mathematical modelling

Please find the attach file I want to summery it. please pick up the main information, background, brief about method and tool, and the expected results) or you can write it as abstract. and please don’t write Equation 1Mathematical modelling of red palm weevil population dynamics in a fragmented environment and under seasonal variability BackgroundDate palm is regarded as the most important crop in the Arabian Peninsula. The Kingdom of Saudi Arabia is one of the four world major producers of dates and the corresponding business activities form a significant part of the country’s economy [1,2]. Red palm weevil (Rhynchophorus ferrugineus) is a pest insect that infests date palms and eventually kills them, it is regarded as the most important date palm pest in the Middle East [3] bringing significant damage to the date production. For this reason, red palm weevil has been a focus of major efforts of Integrated Pest Management. This includes its monitoring (usually by traps [4]) and control measures, the latter is achieved through chemical spraying, use of pheromone traps for mass trapping, and more recently biological control [5]. The adult beetles of Rhynchophorus ferrugineus are relatively large, ranging between two and four centimetres long. Female beetles lay approximately two hundred eggs. The egg hatches into a larvae that can eventually grow to a length of six to seven centimetres. The total life cycle takes about 7–10 weeks, and hence the species can produce several generations per year. The adult beetles cause relatively insignificant damage through feeding. It is the larva that causes most of the problems. It burrow into the trunk of a palm tree up to a metre long, weakening and eventually killing the host plant [6]. Adult beetles of red palm weevil are excellent fliers, mostly active during the day time. The flight activity is known to depend on the ambient temperature [4]. In their flight, weevil can travel significant distances, up to 30-50 km and occasionally even more in just 24 hours [7,8]. The distribution of flight distances combined across season and sex is well described by a normal distribution [7]. Rationale and objectivesMathematical modelling is a powerful research tool to study the properties of real world ecological and agricultural systems. This project aims to develop a mathematical model of the population dynamics of red palm weevil with the goal to identify efficient control strategies. Date palm farms are the main habitat for the pest. Since they are separated by vast areas of a harsh environment unsuitable to support the weevil population, its habitat can be considered as fragmented [9]. Therefore, we are going to model it as a discrete space, e.g. a lattice, where each node of the lattice represents a date farm. The size of the weevil population at a given farm is described by a single variable, say u. Since red pal weevil produces several populations per year, we propose to describe the local population dynamics (at a given node, i.e. at a given farm) by a nonlinear ODE, i.e. du/dt=f(u), where the growth rate dependence on the population density can be either logistic or to account for the Allee effect, or to exhibit multistability. Because the adult beetles can fly over significant distances [7,8], they are likely to travel easily between different plantations. The weevil population dynamics in each given farm are therefore not independent but coupled, through beetle’s dispersal, into a metapopulation. The dynamics of the metapopulation as a whole is then described by the following system:                        duidt=fiui+k=1Nikuk-ui,                                                            (1)where i=1,…,N and N is the total number of nodes in the system and coefficient ik quantify the strength of coupling between habitats i and k. The dynamics of system (1) is known to depend on the strength of coupling, especially in heterogeneous environment (i.e. where parameters of the growth rate function f(u) are different at each lattice node). In particular, an increase in the coupling strength can result in sudden population outbreaks [10]. Recalling that the weevil is a dangerous pest, the population outbreak should be avoided, and one way to avoid it is to apply chemical pesticides. In terms of model (1), application of pesticides means that, at a certain moment t, the population size at i-th node is decreased by a factor, say qi. We hypothesize that timing of pesticides application is important. In order to model the application of pesticides, we extend system (1) by adding a singular perturbation. Namely, at a given moment t=t1, all variables u1,…,uN are reset to smaller values by multiplying them by a factor i, 0<i<1. Arguably, such singular perturbation can change the timing of the outbreak or prevent it altogether, however the precise effect of this singular perturbation is unknown. Regarding the red palm weevil population dynamics, it is known that its flying abilities experience seasonal variation, with highest ability and hence the largest travel distances observed in the summer [5]. In order to account for this, in our model the strength of coupling between different nodes depends on time. Thus, the population outbreak because of the increase in the coupling is more likely to happen in summer. The overall objective of this project is to model the effect of control measures (through application of pesticides) on the population dynamics of weevil as described by model (1). In mathematical terms, the goal of this project is to make an investigation into the ways how the singular perturbation of the dynamics of system (1) can affect the magnitude and the timing of the population outbreak. Our research hypothesis is that, since the strength of the coupling depends on time, so that there is a critical moment t* such as, in order to avoid the population outbreak, the control measure must be applied earlier than t*. This hypothesis will be investigated using analytical methods of the phase plane analysis [10] and computer simulations. The spatial structure of the environment can be further taken into account by the geometry of the lattice. In the first stage of the project, we consider the baseline model where the lattice is rectangular or square. That will allow to identify the main properties of the corresponding dynamical system, in particular the conditions when the population dynamics becomes synchronized. In the second stage, we will consider the lattice of an irregular shape with the purpose to mimic the spatial structure of the date farms.  

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