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In this work, we develop a novel approximation strategy for building almost periodic sequences in the theory of almost periodic functions. Here, we create a different perspective for the argument of Dirichlet in the theory of numbers and design an integer approximation strategy in this regard. The idea behind the strategy comes from Kronecker's theorem and it is proven that for given an almost periodic function, it is possible to design its corresponding almost periodic sequence. Moreover, we provide two population models in both continuous and discrete cases where almost periodic sequence solutions are designed under suitable circumstances. [ABSTRACT FROM AUTHOR]

In ecology and population biology, logistic equation is widely applied for simulating the population of organisms. By combining the logistic model with the low-density effect called Allee effect, several variations of mathematical expressions have been proposed. The upper half of the work was dedicated to establish a novel equation for highly flexible density effect model with Allee threshold. Allee effect has been rarely observed in microorganisms with asexual reproduction despite of theoretical studies. According to the exploitation ecosystem hypotheses, plants are believed to be insensitive to Allee effect. Taken together, knowledge on the existence of low-density effect in photosynthetic microorganisms is required for redefining the ecological theories emphasizing the photosynthetic organisms as the basis for food chains. Therefore, in the lower half of the present article, we report on the possible Allee effect in photo-autotrophic organisms, namely, green paramecia, and cyanobacteria. Optically monitored growth of green paramecia was shown to be regulated by Allee-like weak low-density effect under photo-autotrophic and photo-heterotrophic conditions. Insensitiveness of wild type cyanobacteria (Synechocystis sp. Strain PCC6803) to low-density effect was confirmed, as consistent with our empirical knowledge. In contrast, a mutant line of PCC6803 impaired with a photosynthesis-related pxcA gene was shown to be sensitive to typical Allee's low-density effect (i.e. this line of cells failed to propagate at low cellular density while cells start logarithmic growth at relatively higher inoculating density). This is the first observation that single-gene mutation in an autotrophic organism alters the sensitivity to Allee effect. [ABSTRACT FROM AUTHOR]

We study time series generated by the parametric family of fractional discrete maps introduced by Wu and Baleanu, presenting an alternative way of introducing these maps. For the values of the parameters that yield chaotic time series, we have studied the Shannon entropy of the degree distribution of the natural and horizontal visibility graphs associated to these series. In these cases, the degree distribution can be fitted with a power law. We have also compared the Shannon entropy and the exponent of the power law fitting for the different values of the fractionary exponent and the scaling factor of the model. Our results illustrate a connection between the fractionary exponent and the scaling factor of the maps, with the respect to the onset of the chaos. [ABSTRACT FROM AUTHOR]

This article describes a method for using the United States Census data to open a differential equations course. The question of finding a model for the United States population data gives students a first experience with creating a model using differential equations, and also understanding derivatives, what they mean, and how to calculate them in the context of real data. This model-building start motivates further exploration in many of the standard differential equations topics: the method of separation of variables, slope fields, autonomous equations, equilibria, and stability. [ABSTRACT FROM AUTHOR]

We share our experience and lessons learned from using Systemic Initiative for Modeling Investigations and Opportunities with Differential Equations (SIMIODE) modeling scenarios in our Differential Equations I class at Middle Tennessee State University. Specific projects with Python codes are presented. Discussions are brought forth on how to "best" teach differential equations with modeling approaches while maintaining the balance with the theory. Python notebooks are attached in the Appendix and available at GitHub. [ABSTRACT FROM AUTHOR]

A great deal of educational literature has focused on problem-based learning (PBL) in mathematics at the primary and secondary level, but arguably there is an even greater need for PBL in college math courses. We present a project centered around the Humans versus Zombies moderated tag game played on the Utah State University campus. We discuss the project in the context of an undergraduate differential equations course and discuss how the project was launched. We highlight examples of students mathematical models along with their verbal and written responses, as well as discussing assessment and student learning. Results are discussed in the context of higher and lower cognition levels as well as mathematical appreciation. [ABSTRACT FROM AUTHOR]

The logistic equation with a periodic or asymptotically periodic treatment has a delay either in the per head growth rate or in the net growth rate. When the treatment is constant over time, there exists at most one supercritical Hopf bifurcation for some critical value of the delay. We provide conditions that guarantee the global stability of the trivial steady state when the treatment is an asymptotically periodic function. For the single delayed model and asymptotically periodic drug administration, these are necessary and sufficient conditions. For the double delayed model, given conditions are only sufficient. Simulations for a pharmacokinetic treatment with various periods of drug administration show that the double delayed model is more sensitive than the single delayed model on drug dosage and on the starting time of treatment. [ABSTRACT FROM AUTHOR]

The Beverton–Holt model is a classical population model which has been considered in the literature for the discrete-time case. Its continuous-time analogue is the well-known logistic model. In this paper, we consider a quantum calculus analogue of the Beverton–Holt equation. We use a recently introduced concept of periodic functions in quantum calculus in order to study the existence of periodic solutions of the Beverton–Holtq-difference equation. Moreover, we present proofs of quantum calculus versions of two so-called Cushing–Henson conjectures. [ABSTRACT FROM AUTHOR]

Following Ackleh et al. (2005), we study the multidimensional discrete-time competitive Beverton–Holt equations with equal interspecific competition coefficients. It is shown that competitive exclusion occurs if only one species has the largest carrying capacity. Otherwise, all the species with the largest carrying capacity coexist. In the former case, the system is globally asymptotically stable. In the latter case, the system has a linear stable manifold. [ABSTRACT FROM PUBLISHER]

One of the simplest predator-prey models that tracks the quantity and the quality of prey is the one proposed by [I. Loladze, Y. Kuang, and J.J. Elser, Stoichiometry in producer-grazer systems: Linking energy flow with element cycling, Bull. Math. Biol. 62 (2000) pp. 1137-1162.] (LKE model). In it, the ratio of two essential chemical elements, carbon to phosphorus, C:P, represents prey quality. However, that model does not explicitly track P neither in the prey nor in the media that supports the prey. Here, we extend the LKE model by mechanistically deriving and accounting for P in both the prey and the media. Bifurcation diagrams and simulations show that our model behaves similarly to the LKE model. However, in the intermediate range of the carrying capacity, especially near the homoclinic bifurcation point for the carrying capacity, quantitative behaviour of our model is different. We analyze positive invariant region and stability of boundary steady states. We show that as the uptake rate of P by producer becomes infinite, LKE models become the limiting case of our model. Furthermore, our model can be readily extended to multiple producers and consumers. [ABSTRACT FROM AUTHOR]

The Cushing-Henson conjectures on time scales are presented and verified. The central part of these conjectures asserts that based on a model using the dynamic Beverton-Holt equation, a periodic environment is deleterious for the population. The proof technique is as follows. First, the Beverton-Holt equation is identified as a logistic dynamic equation. The usual substitution transforms this equation into a linear equation. Then the proof is completed using a recently established dynamic version of the generalized Jensen inequality. [ABSTRACT FROM AUTHOR]

In this paper the authors propose a logistic mapping using chaotic model to describe the time-variable pest population. Two kinds of fuzzy rule embedded control strategies are investigated, three segment control and five segment control. They are designed to reduce the pest population. The simulation results show that the objective function is non-convex and anomalous along the control parameters. To find the optimal parameter combinations we develop an ant search approach. By imitating the food hunting and nest moving behaviours of Pachycondyla apicalis ants, this method can adaptively and effectively explore solution areas and arrive at the optimal solution. When we compared the performance curves with the one without control strategy, the method is better and can be used for a wide range of pest control problems in real life. [ABSTRACT FROM AUTHOR]

It appears as if the failure of one organ system precipitates the subsequent failure of other organ systems. The aim of the present analysis is to model such system behavior and understand why medical complications accumulate in individual patients. The human body is first modeled as being comprised of multiple subsystems, with the health of each subsystem dependent on input regarding its own health status and that of all other subsystems. In a second step, the discrete model is generalized into a continuous model that captures system failure, as well as system repair, by a first order differential equation. Failure is approximated by a logistic decline and repair is approximated by a logistic rise in health. A small drop in health of a single subsystem spreads throughout the entire system and affects its overall health. Unless counteracted by measures of therapy or repair, any time-related loss in health of individual subsystems leads to a decline in health of the entire system. The delay in onset of therapy represents the most crucial factor to influence the overall cumulative decline in health. The model suggests that medical management needs to be expeditious to minimize the cumulative time-dependent toll of illness on the entire body. [ABSTRACT FROM AUTHOR]