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This article is concerned with the optimal harvesting of a predator–prey model with a prey refuge and imprecise biological parameters. We consider the model under impreciseness and introduce a parametric functional form of an interval which differs from those of models with precise biological parameters. The existence of all possible equilibria and stability of system are discussed. The bionomic equilibrium of the model is analyzed. Also, the optimal harvesting policy is derived using Pontryagin’s maximal principle. Numerical simulations are presented to verify the feasibilities of our analytical results.

Abstract: Successful integrated pest management (IPM) control programmes depend on many factors which include host–parasitoid ratios, starting densities, timings of parasitoid releases, dosages and timings of insecticide applications and levels of host-feeding and parasitism. Mathematical models can help us to clarify and predict the effects of such factors on the stability of host–parasitoid systems, which we illustrate here by extending the classical continuous and discrete host–parasitoid models to include an IPM control programme. The results indicate that one of three control methods can maintain the host level below the economic threshold (ET) in relation to different ET levels, initial densities of host and parasitoid populations and host–parasitoid ratios. The effects of host intrinsic growth rate and parasitoid searching efficiency on host mean outbreak period can be calculated numerically from the models presented. The instantaneous pest killing rate of an insecticide application is also estimated from the models. The results imply that the modelling methods described can help in the design of appropriate control strategies and assist management decision-making. The results also indicate that a high initial density of parasitoids (such as in inundative releases) and high parasitoid inter-generational survival rates will lead to more frequent host outbreaks and, therefore, greater economic damage. The biological implications of this counter intuitive result are discussed. [Copyright &y& Elsevier]

Abstract. A state-dependent impulsive model is proposed for integrated pest management (IPM). IPM involves combining biological, mechanical, and chemical tactics to reduce pest numbers to tolerable levels after a pest population has reached its economic threshold (ET). The complete expression of an orbitally asymptotically stable periodic solution to the model with a maximum value no larger than the given ET is presented, the existence of which implies that pests can be controlled at or below their ET levels. We also prove that there is no periodic solution with order larger than or equal to three, except for one special case, by using the properties of the LambertW function and Poincaré map. Moreover, we show that the existence of an order two periodic solution implies the existence of an order one periodic solution. Various positive invariant sets and attractors of this impulsive semi-dynamical system are described and discussed. In particular, several horseshoe-like attractors, whose interiors can simultaneously contain stable order 1 periodic solutions and order 2 periodic solutions, are found and the interior structure of the horseshoe-like attractors is discussed. Finally, the largest invariant set and the sufficient conditions which guarantee the global orbital and asymptotic stability of the order 1 periodic solution in the meaningful domain for the system are given using the Lyapunov function. Our results show that, in theory, a pest can be controlled such that its population size is no larger than its ET by applying effects impulsively once, twice, or at most, a finite number of times, or according to a periodic regime. Moreover, our theoretical work suggests how IPM strategies could be used to alter the levels of the ET in the farmers’ favour.

Random noise pervades ecosystems and has the potential for having major impacts on the growth processes of pest populations. In this paper, we aim to investigate the impact of random perturbations on the hormesis Ricker model by considering control measures applied within each generation which can generate hormetic effects, where randomness is characterized by a uniform discrete distribution and white noise, respectively. The main results indicate that the addition of discrete randomness will make the model appear with blurred orbits when the intrinsic growth rate is large enough. The position of the random variable, at the end of each generation or within each generation, cause the blurred orbits to exhibit various forms. Moreover, the effects of variances and expectations of the discrete uniform random variable on the dynamics are evaluated. Further, the introduction of randomness increases the hormetic zone and the maximum response, but can increase or decrease the monotonically increasing interval under different parameter values. In contrast, the stochastic model characterized by white noise, exhibits only a small effect on the bifurcation diagrams with respect to the intrinsic growth rate and the hormesis, which may be attributed to the mean of its noise being zero. Finally, we fit the stochastic model to experimental hormetic data sets observed in multiple fields, and our results demonstrate that the stochastic hormesis Ricker model can capture the characteristics of these data accurately. These findings could provide valuable insights into the understanding of the complexities of pest population dynamics, with implications for better pest control, resource management and for other complex biological systems such as toxicology and drug development. [ABSTRACT FROM AUTHOR]

Abstract. 1. Spectral analysis of 66 years of locust swarm abundance data failed to reveal any significant cycles although the dominant cycle detected in a de‐trended series, adjusted to take account of a significant partial autocorrelation for a lag of 1 year, had a periodicity of 16 years. 2. Although some estimates of the intrinsic rate of increase (r) of desert locusts are indicative of chaos, reconstructions of locust dynamics using response surface methodology (RSM) suggested exponential stability. This was also true for data for the West African Region alone and inclusion of rainfall data from the Sahel improved the significance of an RSM model for West Africa. 3. The observed positive relationship between locust abundance and rainfall in West Africa confirmed the importance of rain; but the variance of the locust abundance also increased with rainfall, making rainfall alone a poor predictor. However, this heteroscedastic pattern was reproducible by a simple logistic model with a chaotic rand a variable K.This was not the case when a stable value for rwas used. 4. The available data and current methodologies are insufficient to provide unequivocal conclusions on locust dynamics, which are complicated by phase changes and associated switches in rvalues.

A recently harvested millet field near Niamey was surveyed for egg-pods of the pest grasshopper Oedaleus senegalensis. The oviposition behaviour, egg-pods and eggs of the species are described. The pods were not randomly distributed and many were laid on the edge of a gentle slope. The density of the pods was 1·14 pods m-2. 142 pods were found of which 13·4% had been attacked by the larvae of Systoechus sp. and 11·3% by other predators. The mean number of eggs in intact pods was 26·4.

Leslie matrices are used to compare theoretical populations of gregarious and solitarious Desert Locusts, Schistocerca gregaria (Forsk.). Despite their lower fecundity, the synchrony and faster maturation of gregarious populations permit them to have much faster rates of increase than solitarious populations. When realistic mortality estimates are assumed the differences can be very pronounced even when the rates of mortality are the same for both phases; this suggests that the longer period during which solitarious locusts are susceptible to predation is critical. The conclusions are briefly discussed with respect to the genesis and maintenance of locust plagues and the evolutionary significance of gregarisation.

A classic susceptible–infected–recovered (SIR) model with nonlinear state-dependent feedback control is proposed and investigated in which integrated control measures, including vaccination, treatment and isolation, are applied once the number of the susceptible population reaches a threshold level. The interventions are density dependent due to limitations on the availability of resources. The existence and global stability of the disease-free periodic solution (DFPS) are addressed, and the threshold condition is provided, which can be used to define the control reproduction number Rc for the model with state-dependent feedback control. The DFPS may also be globally stable even if the basic reproduction number R0 of the SIR model is larger than one. To show that the threshold dynamics are determined by the Rc, we employ bifurcation theories of the discrete one-parameter family of maps, which are determined by the Poincaré map of the proposed model, and the main results indicate that under certain conditions, a stable or unstable interior periodic solution could be generated through transcritical, pitchfork, and backward bifurcations. A biphasic vaccination rate (or threshold level) could result in an inverted U-shape (or U-shape) curve, which reveals some important issues related to disease control and vaccine design in bioengineering including vaccine coverage, efficiency, and vaccine production. Moreover, the nonlinear state-dependent feedback control could result in novel dynamics including various bifurcations.

In this paper, an age-structured infectious disease dynamical model that considers two diseases simultaneously but with limited medical resources is proposed and analyzed. The asymptotic smoothness and persistence of the solution semi-flow are investigated. Then conditions for the existence of a global attractor are derived, which means that disease persists when ℜ 0 > 1. By using a Lyapunov function, it is shown that the infection-free equilibrium is globally asymptotically stable if ℜ 0 < 1 and the infection equilibrium is globally asymptotically stable if ℜ 0 > 1. In the presence of limited medical resources, the results suggest that equitable distribution for the limited medical resources is significant when treating low-risk and high-risk diseases and that keeping a resource sharing coefficient at a moderate level helps to eliminate the disease. • We proposed an age-structured infectious disease model with limited medical resources. • some important parameters were included in ℜ 0. • ℜ 0 determines the global stability of the equilibria. • Discussions of the effects of the parameters and biological significance were provided. [ABSTRACT FROM AUTHOR]

• We developed a stochastic tumour-immune dynamical model concerning pulsed treatment. • Conditions for the three phases of cancer immunoediting are provided. • The results reveal that comprehensive therapy or noise can dominate evolution of tumours. • Biological implications for cancer treatment are presented. Periodical applications of immunotherapy and chemotherapy play significant roles in cancer treatment and studies have shown that the evolution of tumour cells is subject to random events. In order to capture the effects of such noise we developed a stochastic tumour-immune dynamical model with pulsed treatment to describe combinations of immunotherapy with chemotherapy. By using theorems of the impulsive stochastic dynamical equation, the tumour free solution and the global positive solution of the proposed system were investigated. We then show that the expectations of the solutions are bounded. Furthermore, threshold conditions for extinction, non-persistence in the mean, weak persistence and stochastic persistence of tumour cells are provided. The results reveal that comprehensive therapy or noise can dominate the evolution of tumours. Finally, biological implications are addressed and a conclusion is presented. [ABSTRACT FROM AUTHOR]

• We developed a stochastic tumour-immune model with chemotherapeutic dose response. • Conditions for the three phases of cancer immunoediting are provided. • Comprehensive therapy or environmental noise can dominate the evolution of tumours. • We determine a more reasonable treatment for curing cancer. A stochastic tumour-immune dynamical system with pulsed chemotherapeutic dose response is proposed to study how environmental noise affects the evolution of tumours. Firstly, the explicit expression of a tumour-free solution is obtained and then we show that the proposed system exists with a globally asymptotically stable positive solution under certain conditions. Secondly, threshold criteria ensuring the eradication and persistence of tumours are provided. Numerical investigations were carried out to address the effects of key factors on the tumours. The results reveal that environmental noise can dominate all of the tumour dynamics, but comprehensive therapy can not only accelerate the eradication of tumours, but also avoid the disadvantages of a single therapy. [ABSTRACT FROM AUTHOR]

Novel mathematical models with open and closed-loop control for type 1 or type 2 diabetes mellitus were developed to improve understanding of the glucose-insulin regulatory system. A hybrid impulsive glucose-insulin model with different frequencies of glucose infusions and insulin injections was analyzed, and the existence and uniqueness of the positive periodic solution for type 1 diabetes, which is globally asymptotically stable, was studied analytically. Moreover, permanence of the system for type 2 diabetes was demonstrated which showed that the glucose concentration level is uniformly bounded above and below. To investigate how to prevent hyperinsulinemia and hyperglycemia being caused by this system, we developed a model involving periodic intakes of glucose with insulin injections applied only when the blood glucose level reached a given critical glucose threshold. In addition, our numerical analysis revealed that the period, the frequency and the dose of glucose infusions and insulin injections are crucial for insulin therapies, and the results provide clinical strategies for insulin-administration practices. [ABSTRACT FROM AUTHOR]

Pest resistance to pesticides is usually managed by switching between different types of pesticides. The optimal switching time, which depends on the dynamics of the pest population and on the evolution of the pesticide resistance, is critical. Here we address how the dynamic complexity of the pest population, the development of resistance and the spraying frequency of pulsed chemical control affect optimal switching strategies given different control aims. To do this, we developed novel discrete pest population growth models with both impulsive chemical control and the evolution of pesticide resistance. Strong and weak threshold conditions which guarantee the extinction of the pest population, based on the threshold values of the analytical formula for the optimal switching time, were derived. Further, we addressed switching strategies in the light of chosen economic injury levels. Moreover, the effects of the complex dynamical behaviour of the pest population on the pesticide switching times were also studied. The pesticide application period, the evolution of pesticide resistance and the dynamic complexity of the pest population may result in complex outbreak patterns, with consequent effects on the pesticide switching strategies. [ABSTRACT FROM AUTHOR]

Dengue fever is increasing in importance in the tropics and subtropics. Endosymbiotic Wolbachia bacteria as novel control methods can reduce the ability of virus transmission. So, many mosquitoes infected with Wolbachia are released in some countries so that strategies for population replacement can be fulfilled. However, not all of these field trails are successful, for example, releases on Tri Nguyen Island, Vietnam in 2013 failed. Thus, we evaluated a series of relevant issues such as (a) why do some releases fail? (b) What affects the success of population replacement? And (c) Whether or not augmentation can block the dengue diseases in field trials. If not, how we can success be achieved? Models with and without augmentation, incorporating the effects of cytoplasmic incompatibility (CI) and fitness effects are proposed to describe the spread of Wolbachia in mosquito populations. Stability analysis revealed that backward bifurcations and multiple attractors may exist, which indicate that initial quantities of infected and uninfected mosquitoes, augmentation methods (timing, quantity, order and frequency) may affect the success of the strategies. The results show that successful population replacement will rely on selection of suitable strains of Wolbachia and careful design of augmentation methods. [ABSTRACT FROM AUTHOR]

We develop a mathematical model that describes the tumour–immune interaction and the effect on it of pulsed immunotherapy, based on the administration of adoptive cellular immunotherapy (ACI) combined with interleukin-2 (IL-2). The stability conditions for the tumour-free periodic solution are provided with different frequencies of ACI applications and IL-2 infusions. Furthermore, the effects of period, dosage and times of drug deliveries on the amplitudes of the tumour-free periodic solution were investigated. The most feasible immunotherapy strategy was determined by comparing immunotherapy with ACI treatment with or without IL-2. However, to investigate how to enhance the efficacy of chemotherapy (radiotherapy) and reduce its side-effects, we developed a model involving periodic applications of immunotherapy with chemotherapy (radiotherapy) applied only when the density of the tumour reached a given threshold. The results revealed that the initial densities, the effector cell: tumour cell ratios, the periods T and a given critical number of tumour cells C T are crucial for cancer treatment, which confirms that it is important to customize treatment strategies for individual patients. [ABSTRACT FROM AUTHOR]

Abstract: In this paper plant disease models including impulsive cultural control strategies were developed and analyzed. The sufficient conditions under which the infected plant free periodic solution with fixed moments is globally stable are obtained. For the model with an economic threshold (ET) of infected plants, detailed investigations imply that the number of healthy plants either goes to extinction or tends to infinity, and the maximum value of infected plants is always less than the given ET. In order to prevent the healthy plant population going to extinction, we further propose a bi-threshold-value model, which has richer dynamical behavior including order 1-k or order k-1 periodic solutions with . Under certain parameter spaces, the infected plant free periodic solution is globally stable for the bi-threshold-value model. The modeling methods and analytical analysis presented can serve as an integrating measure to identify, evaluate and design appropriate plant disease control strategies. [Copyright &y& Elsevier]

A discrete hormetic Ricker model (HRM) of a population with instant pulse perturbation between two consecutive generations can generate hormetic effects. For example, analysis of complex three-parameter spaces with the intrinsic growth rate, intervention strength, and dose timing as parameters revealed hormetic biphasic dose and dose timing responses. These responses exhibited either a J-shaped or an inverted U-shaped pattern, yielding a homeostatic change or a catastrophic shift and hormetic effects in many parameter regions. However, whether it is pest control or the effectiveness of treatments of tumors with radiotherapy and/or chemotherapy, the phenomenon is linked to the size of the space in which the parameters are located. Thus, the occurrence of hormetic effects is associated with spatiotemporal heterogeneity. To show this, we have developed an integro-difference equation based on the HRM that can describe the complex dynamics and hormetic effects in pest populations in two-dimensional space. Our findings indicate that factors such as the spatial domain, spatial grid, the dosage, and timing of control applications, and the intrinsic growth rate of the pest can significantly affect the spatial and temporal characteristics of pest populations and diverse biological phenomena in two-dimensional space. In particular, under the same size of spatial domains (or the same number of spatial grids), the smaller the spatial grids (or the larger the spatial domain), the stronger the hormetic effects of low dose stimulation and high dose inhibition will be. These factors include larger maximum responses and higher toxic thresholds. Therefore, optimal pest control measures should not only rely on the efficiency of a pesticide and its application time but should also be based on the spatial domain and the designing of reasonable grid-based control strategies to avoid hormetic effects. • A two-dimensional spatial diffusion discrete HRM is proposed using integro-difference equation. • The dynamics of complex three-parameter spaces reveal hormetic biphasic dose and dose timing responses. • The hormetic effects are investigated in the spatial homogeneous context or heterogeneous context. • The experimental designs and analysis can be applied to pest control strategies and tumor treatment. [ABSTRACT FROM AUTHOR]

During the implementation of strong non-pharmaceutical interventions (NPIs), more than one hundred COVID-19 outbreaks induced by different strains in China were dynamically cleared in about 40 days, which presented the characteristics of small scale clustered outbreaks with low peak levels. To address how did randomness affect the dynamic clearing process, we derived an iterative stochastic difference equation for the number of newly reported cases based on the classical stochastic SIR model and calculate the stochastic control reproduction number (SCRN). Further, by employing the Bayesian technique, the change points of SCRNs have been estimated, which is an important prerequisite for determining the lengths of the exponential growth and decline phases. To reveal the influence of randomness on the dynamic zeroing process, we calculated the explicit expression of the mean first passage time (MFPT) during the decreasing phase using the relevant theory of first passage time (FPT), and the main results indicate that random noise can accelerate the dynamic zeroing process. This demonstrates that powerful NPI measures can rapidly reduce the number of infected people during the exponential decline phase, and enhanced randomness is conducive to dynamic zeroing, i.e. the greater the random noise, the shorter the average clearing time is. To confirm this, we chose 26 COVID-19 outbreaks in various provinces in China and fitted the data by estimating the parameters and change points. We then calculated the MFPTs, which were consistent with the actual duration of dynamic zeroing interventions. • The model is proposed to characterize the small scale clustered outbreaks. • The reproduction number change points of multi-waves have been estimated. • Random noise can accelerate the dynamic zeroing process. • NPI measures and implementation time points are crucial for disease control. [ABSTRACT FROM AUTHOR]

The introduction of endosymbiont Wolbachia into laboratory-reared mosquito populations, which are then released to mix with natural populations to prevent the mosquito vectors from reproducing and thus break the transmission cycle of dengue disease, is an innovative new technology. Field trials of Wolbachia -carrying mosquitoes have now been implemented in many countries where there have been the outbreaks of dengue disease. A mathematical model is proposed to investigate the effects of non-identical sex ratio releases of Wolbachia -carrying mosquitoes on the control of dengue transmission. Firstly, we analyzed the existence and stability of equilibria for the system and proved the existence of forward and backward bifurcations. Secondly, bifurcation diagrams, the basins of attraction of the equilibria and the effects of mosquito augmentation for the system with imperfect and perfect transmission rates were obtained. Thirdly, three possible results for mosquito augmentation were summarized for different parameter regions. Further we explored an uncertainty and sensitivity analysis of solutions to estimate the effects of different parameter values on the success or failure of population replacement. Based on the above analysis, we considered a series of relevant issues such as (a) whether or not mosquito augmentation can ensure the success of population replacement? (b) If not, what are the parameter regions for the success or possible success of population replacement? (c) How does the initial density of natural mosquitoes and the quantity of mosquito augmentations affect the success of population replacement? (d) Whether all population replacements are effective for reducing the spread of dengue virus in the end? The results of this study will be helpful for public health authorities in designing proper strategies of mosquito augmentations for the control of dengue disease. [ABSTRACT FROM AUTHOR]

Modeling external perturbations such as chemical control within each generation of discrete populations is challenging. Based on a method proposed in the literature, we have extended a discrete single species model with multiple instantaneous pesticide applications within each generation, and then discuss the existence and stability of the unique positive equilibrium. Further, the effects of the timing of pesticide applications and the instantaneous killing rate on the equilibrium were investigated in more detail and we obtained some interesting results, including a paradox and the cumulative effects of the incorrect use of pesticides on pest outbreaks. In order to show the occurrences of the paradox and of hormesis, several special models have been extended and studied. Further, the biological implications of the main results regarding successful pest control are discussed. All of the results obtained confirm that the cumulative effects of incorrect use of pesticides may result in more severe pest outbreaks and thus, in order to avoid a paradox in pest control, control strategies need to be designed with care, including decisions on the timing and number of pesticide applications in relation to the effectiveness of the pesticide being used. [ABSTRACT FROM AUTHOR]

Highlights: [•] We develop novel mathematical model with evolution of pest resistance. [•] Three pesticide switching methods have been proposed and analyzed. [•] The optimal pesticide switching strategy has been discussed. [•] When pest managers should switch pesticides can be determined analytically. [ABSTRACT FROM AUTHOR]

• Impulsive differential equations were introduced to model intermittent androgen deprivation therapy. • The effects of white noises and tumour antigenicity on the dynamics of prostate cancer cells were discussed. • Sufficient conditions for the stationary distribution and ergodicity of the system were provided. • Immunotherapy can promote the therapeutic effect of intermittent androgen deprivation therapy. Intermittent androgen deprivation therapy is often used to treat prostate cancer, but there are few mathematical modelling studies of it. To explore the mechanisms of such therapy, we describe intermittent therapy with impulsive differential equations, then we propose a novel mathematical model of intermittent androgen deprivation therapy with white noise. We first studied the model's basic properties including the existence and uniqueness of the solution. By using the theory of stochastic differential equations, we investigated the thresholds for the extinction and persistence of prostate cancer cells, which are markedly affected by the antigenicity of tumours and noise parameters. Moreover, sufficient conditions for the stationary distribution and ergodicity of the system are provided. The results show that reducing the period of pulsed interventions or increasing the dosages (or frequencies) of the therapy will be helpful for curing prostate cancer. [ABSTRACT FROM AUTHOR]