The Existence of Positive Periodic Solutions for a Kind of Delay Logistic Equations

In the present paper, we consider the existence of positive periodic solutions for a kind of delay Logistic equations. By using a fixed point theorem in cones, we give some new existence results of single and multiple positive periodic solutions for a kind of delay Logistic equations. Some biomathematical models are presented to illustrate our results.

Mathematical model for predicting fungal growth and decomposition rates based on improved Logistic equations

As the function of the decomposition of fungi has been clearly researched in the global carbon cycle,it is obviously of value to explore the decomposition rate of fungal populations.This study analyzed the relationship between environmental factors and biodiversity step by step.In order to explore the interaction between the fungi and the relationship between the decomposition rate of fungi with time,the model based on the Logistic model was built and the Lotka-Volterra model was employed in the condition of two kinds of fungi existing in an environment with limited resources.The changing trend of population number and decomposition rate of several fungi under different environmental conditions can be predicted through the model.To illustrate the applicability of the model,Laetiporus conifericola and Hyphoderma setigerum were applied as examples.The results showed that the higher the degree of population diversity,the greater the decomposition rate,and the higher the decomposition efficiency of the ecosystem.Its rich species diversity is conducive to accelerating the decomposition of litter,lignocellulose,and the circulation of the entire ecosystem.Based on the above model and using the data from measuring the mycelial elongation rate of each isolate at 10℃,16℃,and 22℃ under standardized laboratory conditions,the growth patterns of the five fungi combinations were simulated.The results revealed a general increase in growth rate with increasing temperature,which verifies the accuracy of the model.Moreover,it also revealed that the total decomposition rate after fungal incorporation was negatively correlated with the decomposition rate of a fungal single action.Based on the above model,predictions can be made for fungal growth in different environments,and suitable environments for fungal growth can be determined.In the future,the model can be further optimized,and lignin and cellulose decomposition factors can be added to fit the decomposition of logs.The application scenarios of the model can be further broadened,which can contribute to the restoration and management of the ecological environment,as well as produce good effects in the fields of fungi assisting the global carbon cycle and soil problem restoration.

Study on Sex Ratio of Lampreys Based on Simulated Ecosystem-Food Web Model

Lampreys, as an important participant in the ecosystem, play an irreplaceable role in the stability of nature. A variety of models were used to simulate ecosystems and food webs, and the dynamic evolution of multiple populations was solved. The temporal changes of the biomass and the health of the ecosystem affected by the population of Lampreys in other ecological niches were solved. For problem 1, Firstly, a simple natural ecosystem is simulated based on the threshold model and BP neural network model. The dynamic change of the sex ratio of lampreys population and the fluctuation of ecosystem health value were found to generate time series maps. Lampreys overprey on low-niche animals, which damages the overall stability of the ecosystem. For problem 2, We used the Lotka-Volterra model to construct ecological competition between lampreys and primary consumers and predators. Then, the Lotka-Volterra equations were solved, and a control group without gender shift function was set up, which reflected the advantages and disadvantages of the sex-regulated characteristics of lampreys in the natural environment. For problem 3, The ecosystem model established in question 1 was further deepened, and the food web was simulated by the Beverton-Holt model and the Logistic time-dependent differential equations model. The parameters of the food web model were input into the neurons of the ecosystem model, and the two models were integrated to form an overall biosphere model. The output layer of the ecosystem neural network was input into the food web Beverton-Holt and Logistic differential equations, and finally, the three-dimensional analytical solution was obtained by numerical simulation. Then Euler method is used to obtain the exact value of the solution surface. The Random forest model was used to predict the future development of lampreys and other ecological niches. For problem 4, By investigating relevant literature, we normalized the populations of lampreys and a variety of fish as well as other ecological niche animals, plants and microorganisms in the same water area, set different impact weights of lampreys, constructed weight evaluation matrix, and obtained positive and negative ideal solution vectors and negative correlation proximity by using TOPSIS comprehensive evaluation method. It is concluded that many kinds of fish are greatly affected by the sex regulation of lampreys.

A Logistic-growth-equation-based Intensity Prediction Scheme for Western North Pacific Tropical Cyclones

Accurate prediction of tropical cyclone(TC)intensity remains a challenge due to the complex physical processes involved in TC intensity changes.A seven-day TC intensity prediction scheme based on the logistic growth equation(LGE)for the western North Pacific(WNP)has been developed using the observed and reanalysis data.In the LGE,TC intensity change is determined by a growth term and a decay term.These two terms are comprised of four free parameters which include a time-dependent growth rate,a maximum potential intensity(MPI),and two constants.Using 33 years of training samples,optimal predictors are selected first,and then the two constants are determined based on the least square method,forcing the regressed growth rate from the optimal predictors to be as close to the observed as possible.The estimation of the growth rate is further refined based on a step-wise regression(SWR)method and a machine learning(ML)method for the period 1982−2014.Using the LGE-based scheme,a total of 80 TCs during 2015−17 are used to make independent forecasts.Results show that the root mean square errors of the LGE-based scheme are much smaller than those of the official intensity forecasts from the China Meteorological Administration(CMA),especially for TCs in the coastal regions of East Asia.Moreover,the scheme based on ML demonstrates better forecast skill than that based on SWR.The new prediction scheme offers strong potential for both improving the forecasts for rapid intensification and weakening of TCs as well as for extending the 5-day forecasts currently issued by the CMA to 7-day forecasts.

The Existence of Periodic and Almost Periodic Solutions of Logistic Type Equation

In this paper, we study the Logistic type equation x= a（t）x -b（t）x^2＋ e（t）. Under the assumptions that e（t） is small enough and a（t）, b（t） are contained in some positive intervals, we prove that this equation has a positive bounded solution which is stable. Moreover, this solution is a periodic solution if a（t）, b（t） and e（t） are periodic functions, and this solution is an almost periodic solution if a（t）, b（t） and e（t） are almost periodic functions.

Oscillation in a Variable Delay Logistic Difference Equation

Consider the nonautonomous delay logistic equation △yn=pnyn（1-yn-ln/k）,n≥0, where {Pn}n≥0 is a sequence of nonnegative real numbers, {In}n≥0 is a sequence of positive integers satisfying n→∞lim（n-ln）=∞, and k is a positive constant. Only solutions which are positive for n ≥ 0 are considered. We obtain a new sufficient for all positive solutions of （1） to oscillate about k which contains the corresponding result in [2] when i = 1.

GLOBAL ATTRACTIVITY IN A DELAY LOGISTIC DIFFERENCE EQUATION

This paper studies the global attractivity of the positive equilibrium 1 of the delay logistic difference equation Δ y n=p ny n(1-y τ(n) ), n=0,1,2,...,(*)where {p n} is a sequence of positive real numbers, {τ(n)} is a nondecreasing sequence of integers, τ(n)<n and lim n→∞τ(n)=∞ .It is proved that ifnj=τ(n)p j≤54 for sufficiently large n and ∞j=0p j=∞,then all positive solutions of Eq.(*) tend to 1 as n→∞ .The results improve the existing results in literature.

Existence of positive solutions in a delay logistic difference equation

The author studied the existence of positive solutions of the delay logistic difference equation Δ y n=p ny n(1-y τ(n) ), n =0,1,2,.... where { p n } is a sequence of positive real numbers, { τ(n) } is a nondecreasing sequence of integers, τ(n)<n and lim n →∞ τ(n) =∞. A sufficient condition for the existence of positive solutions of the equation was given.

Differential Calculus the Study of the Growth and Decay of an Entity’s Population

Population Growth and Decay study of the growth or the decrease of a population of a given entity, is carried out according to the environment. In an infinite environment, i.e. when the resources are unlimited, a population P believes according to the following differential equation P’ = KP, with the application of the differential calculus we obtasin an exponential function of the variable time (t). The function of which we can predict approximately a population according to the signs of k and time (t). If k > 0, we speak of the Malthusian croissant. On the other hand, in a finite environment i.e. when resources are limited, the population cannot exceed a certain value. and it satisfies the logistic equation proposed by the economist Francois Verhulst: P’ = P(1-P).

Qualitative behavior of discrete-time Caputo-Fabrizio logistic model with Allee effect

The aim of this paper is to investigate the dynamic behaviors of fractional-order logistic model with Allee effects in Caputo-Fabrizio sense.First of all,we apply the two-step Adams-Bashforth scheme to discretize the fractional-order logistic differential equation and obtain the two-dimensional discrete system.The parametric conditions for local asymptotic stability of equilibrium points are obtained by Schur-Chon criterion.Moreover,we discuss the existence and direction for Neimark-Sacker bifurcations with the help of center manifold theorem and bifurcation theory.Numerical simulations are provided to illustrate theoretical discussion.It is observed that Allee effect plays an important role in stability analysis.Strong Allee effect in population enhances the stability of the coexisting steady state.In additional,the effect of fractional-order derivative on dynamic behavior of the system is also investigated.

Use of New Water Soluble Surface Film－Forming Material to Reduce Ammonia Loss from Water Solution

A new water soluble surfaCe film-forming material was developed and its effect on reducing ammonia volatilization from an alkaline solution was investigated in laborstory. Results showed that the new film formed by the material was not only more effective in reducing ammonia loss than any other films tested but also much cheaper. The optimum amount of addition of the new film-forming material was about 10times the theoretical amount to form a monomolecular film. Under the experimental conditions, the new film could effectively depress the ammonia volatilization for at least 6 days. The cumulative ammonia loss rates for different films were fitted to a simple logistic equation, and some important parameters such as the cumulative loss, and the maximum and average volatilization rates were calculated. The effect of different films could be, therefore, compared quantitatively, indicating the new film was most effective in depressing ammonia volatilization.

A numerical study of coupled maps representing energy exchange processes between two environmental interfaces regarded as biophysical complex systems

The field of environmental sciences is abundant with various interfaces and is the right place for the application of new fundamental approaches leading towards a better understanding of environmental phenomena. Following the definition of environmental interface by Mihailovic and Bala? [1], such interface can be, for example, placed between: human or animal bodies and surrounding air, aquatic species and water and air around them, and natural or artificially built surfaces (vegetation, ice, snow, barren soil, water, urban communities) and the atmosphere, cells and surrounding environment, etc. Complex environmental interface systems are (i) open and hierarchically organised (ii) interactions between their constituent parts are nonlinear, and (iii) their interaction with the surrounding environment is noisy. These systems are therefore very sensitive to initial conditions, deterministic external perturbations and random fluctuations always present in nature. The study of noisy non-equilibrium processes is fundamental for modelling the dynamics of environmental interface regarded as biophysical complex system and for understanding the mechanisms of spatio-temporal pattern formation in contemporary environmental sciences. In this paper we will investigate an aspect of dynamics of energy flow based on the energy balance equation. The energy exchange between interacting environmen- tal interfaces regarded as biophysical complex systems can be represented by coupled maps. Therefore, we will numerically investigate coupled maps representing that exchange. In ana- lysis of behaviour of these maps we applied Lyapunov exponent and cross sample entropy.

Exact multiplicity of solutions to perturbed logistic type equations on a symmetric domain

We apply the imperfect bifurcation theory in Banach spaces to study the exact multiplicity of solutions to a perturbed logistic type equations on a symmetric spatial domain. We obtain the precise bifurcation diagrams.

From Pressure-Volume Relationship to Volume-Energy Relationship: A Thermo-Statistical Model for Alveolar Micromechanics

The connective tissue fiber system and the surfactant system are essential and interdependent components of lung elasticity. Despite considerable efforts over the last decades, we are still far from understanding the quantitative roles of either the connective tissue fiber or the surfactant systems. Through thermo-statistic considerations of alveolar micromechanics, the author introduced a thermo-statistic state function “entropy” to analyze the elastic property of pulmonary parenchyma based on the origami model of alveolar polyhedron. By use of the entropy for alveolar micromechanics, from the logistic equation for the static pressure (P)-volume (V) curves including parameters a, b, c, and k (V - a = b/[1+ exp{-k (P - c)}]), a set of equations was obtained to define the internal energy of lungs (UL) and its corresponding lung volume (VL). Then, by use of parameters a, b, c, and k, an individual volume-internal energy (VL - UL) diagram was constructed from reported data in patients on mechanical ventilation. Each VL - UL diagram constructed was discussed that its minimal value Uo = c (a + b/2) and its shape parameter b/k represent quantitatively the energy of tissue force and the energy of surface force. Furthermore, by use of the VL - UL relationship, the hysteresis of lungs estimated by entropy production was discussed as dependent on the difference in the number of contributing pulmonary lobules. That is, entropy production might be a novel quantitative indicator to estimate the dynamics of the bronchial tree. These values obtained by combinations of parameters of the logistic P-V curve seem useful indicators to optimize setting a mechanical ventilator. Thus, it is necessary to develop easy tools for fitting the individual sigmoid pressure-volume curve measured in the intensive care unit to the logistic equation.

FURTHER IN VESTIGATION OF THE VOLTERRA-LOTKA COMPETITION MODEL WITH DIFFUSION

In this paper, the existence of coexistence states in the Voterra Lotka competition model under Dirichlet boundary conditions is the major concern. Attention will be focused on the logistic equation. With a comparison theorem an inequality satisfied by the solution of logistic equation is established. Some improved sufficient conditions for the existence of coexistence states are obtained.

ON A PERIODIC DELAY LOGISTIC TYPE POPULATION MODEL

In this paper, we study the existence of a positive periodic solution forN′(t)=r(t)1-x(t-τ(t)K(t) θ ,and obtain sufficient conditions for the periodic solution to be globally attractive.

Existence,uniqueness,and global attractivity of positive solutions and MLE of the parameters to the Logistic equation with random perturbation

This paper discusses a randomized Logistic equation $\dot N(t) = (r + \alpha \dot B(t))N(t)[1 - \frac{{N(t)}}{K}]$ with an initial value N(0) = N 0, and N 0 is a random variable satisfying 0 < N 0 < K. The existence, uniqueness and global attractivity of positive solutions and maximum likelihood estimate (MLE) of the parameters of the equation are studied.

Dynamic Behavior of a Logistic Type Equation with Infinite Delay

A non-autonomous Logistic type equation with infinite delay is investigated. For general nonautonomous case, sufficient conditions which guarantee the uniform persistence and globally attractivity of the system arc obtained; For almost periodic case, by means of a suitable Lyapunov functional, sufficient conditions are derived for the existence and uniqueness of almost periodic solution of the system. Some new results are obtained.

NECESSARY AND SUFFICIENT CONDITIONS FOR THE OSCILLATION OF A DELAY LOGISTIC EQUATION WITH CONTINUOUS AND PIECEWISE CONSTANT ARGUMENTS

In this paper, we obtain some necessary and sufficient conditions for the oscillation of all positive solutions of a delay Logistic equation with continuous and piecewise constant arguments about the positive equilibrium.