Abstract:Due to the fear to predation risk preys may decrease birth rate or flee high level of predators patch to prey-only patch at a cost of decreased resources. In this paper a predator-prey model with fear effect and patch structure is constructed to study how the fear and diffusion affect predator-prey dynamics. The stability of equilibria and existence of Hopf bifurcation are studied. The fear to predation risk can be modeled by two types of parameters: $k$, $a_{21}$ and $a_{12}$, where high level of fear leads to large $k$ and thus low level of prey's birth rate; high level of fear results in large prey's diffusive rate $a_{21}$ from predator-prey patch 1 to prey-only patch 2, and great hunger and bad ability of remembering fear in patch 1 cause large prey's diffusive rate $a_{12}$ from patch 2 to patch 1. Numerical simulations are as follows. (1) In some cases, large $k$ can stabilize the predator-prey system by excluding the existence of periodic solutions when $a_{21}$ is small. However, when $a_{21}$ is large the change of $k$ can not lead to periodic oscillations. In addition, when $a_{21}$ is larger, the predators will die out. Thus, the oscillation behavior or the persistence of predators may be overestimated if the diffusive behaviors $a_{21}$ is weakened or ignored. (2) Under some situations, the change of $a_{12}$ causes Hopf bifurcations twice. This implies that the oscillation behavior may be underestimated or overestimated if the diffusive behavior $a_{12}$ is weakened. %Numerical simulations show that high levels of fear (or low birth rate of preys) can stabilize the predator-prey system by excluding the existence of periodic solutions. However, high level of dispersal caused by fear from predator-prey patch to predator-free patch can stabilize the predator-prey system. On the contrary, high level of dispersal caused by starvation from predator-free patch to predator-prey patch can induce periodic oscillations. % These conclusions imply that the oscillation behavior may be underestimated or overestimated according to whether the fear effect to predation risk (or low birth rate of preys) or dispersal caused by starvation dominates if the dispersal is ignored.

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	1. Dynamics of a Predator-Prey Model with Fear Effect and Patch Structure
	LAN Zi-Teng,ZHANG Yu-Wei,WEN Luo-Sheng,ZHANG Tian-Ran
	Mathematics 16 March 2024
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	Abstract:Due to the fear to predation risk preys may decrease birth rate or flee high level of predators patch to prey-only patch at a cost of decreased resources. In this paper a predator-prey model with fear effect and patch structure is constructed to study how the fear and diffusion affect predator-prey dynamics. The stability of equilibria and existence of Hopf bifurcation are studied. The fear to predation risk can be modeled by two types of parameters: $k$, $a_{21}$ and $a_{12}$, where high level of fear leads to large $k$ and thus low level of prey's birth rate; high level of fear results in large prey's diffusive rate $a_{21}$ from predator-prey patch 1 to prey-only patch 2, and great hunger and bad ability of remembering fear in patch 1 cause large prey's diffusive rate $a_{12}$ from patch 2 to patch 1. Numerical simulations are as follows. (1) In some cases, large $k$ can stabilize the predator-prey system by excluding the existence of periodic solutions when $a_{21}$ is small. However, when $a_{21}$ is large the change of $k$ can not lead to periodic oscillations. In addition, when $a_{21}$ is larger, the predators will die out. Thus, the oscillation behavior or the persistence of predators may be overestimated if the diffusive behaviors $a_{21}$ is weakened or ignored. (2) Under some situations, the change of $a_{12}$ causes Hopf bifurcations twice. This implies that the oscillation behavior may be underestimated or overestimated if the diffusive behavior $a_{12}$ is weakened. %Numerical simulations show that high levels of fear (or low birth rate of preys) can stabilize the predator-prey system by excluding the existence of periodic solutions. However, high level of dispersal caused by fear from predator-prey patch to predator-free patch can stabilize the predator-prey system. On the contrary, high level of dispersal caused by starvation from predator-free patch to predator-prey patch can induce periodic oscillations. % These conclusions imply that the oscillation behavior may be underestimated or overestimated according to whether the fear effect to predation risk (or low birth rate of preys) or dispersal caused by starvation dominates if the dispersal is ignored.
	TO cite this article:LAN Zi-Teng,ZHANG Yu-Wei,WEN Luo-Sheng, et al. Dynamics of a Predator-Prey Model with Fear Effect and Patch Structure[OL].[16 March 2024] http://en.paper.edu.cn/en_releasepaper/content/4762503


	2. Dynamics and control of malaria transmission model with vaccination and patch structure
	ZENG Si-Jia, ZHANG Tian-Ran, ZHANG Tian-Ran
	Mathematics 15 March 2024
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	Abstract:Malaria is an infectious disease transmitted by mosquitoes, and this paper focuses on two diﬀerent regions due to diﬀerences in prevention and control measures such as vaccination. When a movement of population and mosquitoes occurs between two patches, under what conditions will malaria eventually extinct or continue to spread. This article takes the basic regeneration number as the threshold parameter. When $\mathscr{R}_0< 1$ the disease will die out and when $\mathscr{R}_0> 1$ the disease will persist. In section 5, through numerical simulation, we found that rational distribution of vaccine number between two patches can minimize $\mathscr{R}_0$ and minimize the total number of infections when the number of vaccine is limited. The results show that the distribution of vaccine number is diﬀerent from the conventional idea that the distribution is based on the patches population ratio. Due to the inﬂuence of population movement between patches, the optimal strategy for vaccine distribution needs to be based on the actual situation.
	TO cite this article:ZENG Si-Jia, ZHANG Tian-Ran, ZHANG Tian-Ran. Dynamics and control of malaria transmission model with vaccination and patch structure[OL].[15 March 2024] http://en.paper.edu.cn/en_releasepaper/content/4762506

Saved Papers

Saved Papers


	3. N-soliton solutions and nonlinear dynamics for a generalized Broer–Kaup system
	LIU Tian-Zhi,JIANG Yan,TIAN Bo,BAI Fan
	Mathematics 10 February 2023
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	Abstract:Water waves are observed in many situations, as well as on rivers, lakes or oceans. Under consideration in this paper is a famous dispersion water wave model: the (1 + 1)-dimensional generalized Broer–Kaup (gBK) system. This system is used to simulate the bi-directional propagation of long waves in shallow water. Based on the bilinear forms given in this paper, novel N-soliton solutions of this gBK system is obtained by using Hirota's bilinear method. In order to understand the nonlinear dynamics localized in the gBK systems, local structures of the obtained one-, two-, three- and four-soliton solutions are shown. This paper reveals the local structures of the one-soliton solutions and interactions between multi-soliton solutions and preliminarily explains the nonlinear dynamical characteristics of bell soliton and kink soliton in this gBK system.
	TO cite this article:LIU Tian-Zhi,JIANG Yan,TIAN Bo, et al. N-soliton solutions and nonlinear dynamics for a generalized Broer–Kaup system[OL].[10 February 2023] http://en.paper.edu.cn/en_releasepaper/content/4759074


	4. Solitons, breathers and lumps for a generalized (3+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona-Mahony equation in a fluid
	BAI Fan,BAI Fan, JIANG Yan, JIANG Yan,TIAN Bo,TIAN Bo,LIU Tian-Zhi,LIU Tian-Zhi
	Mathematics 08 January 2023
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	Abstract:Fluids are studied in such disciplines as atmospheric science, oceanography and astrophysics. In this paper, we investigate a generalized (3+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona-Mahony equation in a fluid. Via the Hirota method, bilinear forms, soliton, breather and lump solutions of that equation are obtained. At the same time, solitons, breathers and lumps are depicted. We find that the amplitude and shape of the one soliton keep unchanged during the propagation and the interaction between the two solitons is elastic. We observe that the shapes and amplitudes of the breather and lump remain unchanged during the propagation. We present the one solitons, two solitons, breathers and lumps with the influence of the coefficients in the equation.
	TO cite this article:BAI Fan,BAI Fan, JIANG Yan, et al. Solitons, breathers and lumps for a generalized (3+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona-Mahony equation in a fluid[OL].[ 8 January 2023] http://en.paper.edu.cn/en_releasepaper/content/4758823


	5. Multi-attribute three-way decision making with degree-based linguistic term sets
	Wang Zeng-Hui,Zhu Ping
	Mathematics 11 June 2021
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	Abstract:When using several possible linguistic terms with different weights (represented by probability values) to express preferences, the probability distribution of those linguistic terms is usually hard to be obtained. In this paper, we utilize degree values to reflect the weights of possible linguistic terms and first propose a novel concept called degree-based linguistic term set (DLTS) from the perspective of degree. Then, we combine multi-attribute decision making and three-way decision to present a multi-attribute three-way decision method under the environment of DLTSs. Finally, we illustrate the validity of our decision method through a concrete example.
	TO cite this article:Wang Zeng-Hui,Zhu Ping. Multi-attribute three-way decision making with degree-based linguistic term sets[OL].[11 June 2021] http://en.paper.edu.cn/en_releasepaper/content/4755218


	6. Extinction Time Analysis under cyclic-dominant stochastic population
	ZHANG Li-Bin, WU Bin
	Mathematics 19 March 2021
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	Abstract:In multi-species population, the extinction event is one of the most important phenomena during the evolution of the population.The fate of one species not only determined by itself, but also influenced by other species with which it interacts.Therefore, the extinction of one species has huge influenced on the whole population.Previous researches shows how randomness affects the extinction time of finite population by simulation.However, the mathematical analysis of extinction time is absent.Here we investigate the extinction time under the finite population which adopts the cyclic Rock-Paper-Scissors game.We find a solution of extinction time based on stochastic process under neutral drift.Then for the situation that selection intensity is not vanishing, we get a partial differential equation (PDE) to depict the extinction time.We analyse the PDE under weak selection intensity.Moreover, although our research starts from the model adopting the Fermi process, and we proved that the extinction time under the general imitation process can be known based on our results of the Fermi process.
	TO cite this article:ZHANG Li-Bin, WU Bin. Extinction Time Analysis under cyclic-dominant stochastic population[OL].[19 March 2021] http://en.paper.edu.cn/en_releasepaper/content/4754233


	7. Finite-time Synchronization for coupled nonidentical Duffing-type oscillatordynamical networks
	LV Mengting,ZHANG Zhengqiu,REN Ling
	Mathematics 08 March 2021
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	Abstract:In this paper, the finite-time synchronization for a class of coupled nonidentical drive-response Duffing-type oscillator dynamical networks is studied. using differential analytical techniques from exisiting papers, and establishing driving response system model of coupling Duffing dynamic network . Then the error system was obtained by transforming the driving response system and designed a reasonable controller , two novel conditions toassure the finite-time synchronization for above networks are acquired by combining integral inequality skills with transforming two differential inequalities into a differential inequality. Based on the results given in this paper, two examples are listed, which implies the validity of these conditions Meanwhile, getting rid of utilizing the finite-time synchronization theorems, our study work widens the research method of the finite-time synchronization for dynamical networks.
	TO cite this article:LV Mengting,ZHANG Zhengqiu,REN Ling. Finite-time Synchronization for coupled nonidentical Duffing-type oscillatordynamical networks[OL].[ 8 March 2021] http://en.paper.edu.cn/en_releasepaper/content/4753943


	8. Disaster Response System Based on GIS
	JIAO Yanan,AI Gang,ZHANG Kehui,WANG Yue
	Mathematics 09 September 2019
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	Abstract:Puerto Rico was hit by a hurricane in 2017. In this paper, we will design a disaster response system for help, inc., which makes full use of the various devices and containers mentioned in this paper, and can meet the anticipated medical needs in the event of future disasters. According to the number of medical packages received during the duration of the hurricane on the island, the number of cargo holds that received medical packages was calculated. Through the analysis of given information, the carrying capacity, photographic capability and loading capacity of UAV can be obtained. The different attributes of UAV are given different weights by analytic hierarchy process, and the performance of different UAV is obtained. The packing model is used to calculate the optimal components in the cargo hold. Then, an optimization model is established to calculate and minimize the number of UAVs. Then choose the place where ISO cargo containers will be placed. According to the location of the port and the point of demand, two containers were allocated to distribute medical kits for the Pavia Arecibo Hospital and Caribbean Medical Center. The location of three containers is selected by constructing Tyson polygon. Unmanned aerial vehicles in ISO containers will complete photographic missions in the corresponding areas. Finally, the expression of flight path is obtained by depth-first traversal of binary tree, and the flight schedule is obtained according to the length of road and the flight speed of UAV.
	TO cite this article:JIAO Yanan,AI Gang,ZHANG Kehui, et al. Disaster Response System Based on GIS[OL].[ 9 September 2019] http://en.paper.edu.cn/en_releasepaper/content/4749616


	9. Existence of solusions for indefinite linear Choquard equations with Hardy-Littlewood-Sobolev critical exponents
	WANG Ling,WANG Fei-Zhi
	Mathematics 27 May 2019
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	Abstract:This paper are concerned with the following linear Choquard equation\begin{equation} -\Delta u +V(x)u =K(x)(\int_{\mathbb{R}^{N}}\frac{\|u\|^{2^{}_{\mu}}}{\|x-y\|^{\mu}}dy)\|u\|^{2^{}_{\mu}-2}u + g(x,u) \quad x \in \mathbb{R}^{N},\end{equation}where $N \geq 3$, $0<\mu<N$ and $ 2^{}_{\mu}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. V(x) is a continuous function such that the spectrum $\sigma(-\Delta+V(x))$ of $-\Delta+V(x)$ in $L^{2}(R^{N})$ has a negative part, K(x) is a bounded positive function, g is of subcritical growth. The existence of nontrivial solutions has been obtained by variational methods.ssume the following hypotheses:$ $\\$(V_{1})$: $V(x)\in C(\mathbb{R}^{N})\bigcap L^{\infty}(\mathbb{R}^{N})$ and $\liminf_{\| x \|\rightarrow \infty} V(x)=v_{\infty} >0.$\\$(V_{2})$: $(W_{1}(x)-v_{\infty}) \in L^{N/2}(\mathbb{R}^{N})$, $0\notin \sigma(-\Delta+V)$ and $\sigma(-\Delta+V)\bigcap (-\infty,0)\neq \emptyset$,$\quad$ where $\sigma$ denotes the spectrum in $L^{2}(\mathbb{R}^{N})$ and $W_{1}(x)=max\{V(x),v_{\infty}\}.$\\$(K_{1})$: $K(x)\in C(\mathbb{R}^{N})$ attains its maximum at 0. $K_{M}:=K(0)=\max_{\mathbb{R}^{N}}K(x)$ and$\quad$ there exist positive constants $K_{min}$ and $\alpha$ such that $K(x)\geq K_{min}$ and $K(0)$$\quad$ $-K(x)=O(\|x\|^{\alpha})$.\\$(G_{1})$: $g\in C(\mathbb{R}^{N}\times\mathbb{R}, \mathbb{R})$ and $\|g(x,s)\| \leq \omega(x)\|s\|+h(x)\|s\|^{p-1}$, where $\omega(x)$$\in L^{N/2}(\mathbb{R}^{N})$$\quad$ $\bigcap L^{\infty}(\mathbb{R}^{N})$, $2<p<$ $2^{}$ and $h(x)\in L^{\frac{2^{}}{2^{}-P}}(\mathbb{R}^{N})\bigcap L^{\infty}(\mathbb{R}^{N}).$\\$(G_{2})$: $\lim_{s\rightarrow 0}g(x,s)/s=0$ uniformly on $\mathbb{R}^{N}.$\\$(G_{3})$: $0\leq 2G(x,s)\leq sg(x,s)$ for a.e. $x \in \mathbb{R}^{N}, \forall s \in \mathbb{R},$ where $G(x,s):=\int_{0}^{s}g(x,t)dt.$$ $
	TO cite this article:WANG Ling,WANG Fei-Zhi. Existence of solusions for indefinite linear Choquard equations with Hardy-Littlewood-Sobolev critical exponents[OL].[27 May 2019] http://en.paper.edu.cn/en_releasepaper/content/4749042


	10. Effect of seasonal changing temperature on the growth of phytoplankton
	CHEN Ming,FAN Meng,YUAN Xing,ZHU Huai-Ping
	Mathematics 06 May 2017
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	Abstract:A non-autonomous nutrient-phytoplankton interacting model incorporating the effect of time-varying temperature is established. The impacts of temperature on metabolism of phytoplankton such as nutrient uptake, death rate, and nutrient releasing from particulate nutrient are investigated. The ecological reproductive index is formulated to present a threshold criteria and to characterize the dynamics of phytoplankton.The effect of seasonal temperature and daily temperature on phytoplankton biomass are simulated numerically. Numerical simulation shows that the phytoplankton biomass is very robust to the variation of water temperature. The performance and the adaptability of the model is well assessed by carrying out an application scenario to our field study in Lake Tai in China. The dynamics of the model and model predictions agree well with the experimental data. Our model provides a reasonable explanation of triggering mechanism of phytoplankton bloom.
	TO cite this article:CHEN Ming,FAN Meng,YUAN Xing, et al. Effect of seasonal changing temperature on the growth of phytoplankton[OL].[ 6 May 2017] http://en.paper.edu.cn/en_releasepaper/content/4729441

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