Population Growth Equation Research Articles

Population growth equation by fuzzy common integral transforms

In this work, three famous different fuzzy integral transforms have been attempted to evaluate the exact solution of fuzzy equation based on fuzzy convolution Volterra integral equation of the second kind due to the fact that such transforms reduce the integral problem to algebraic problem. Moreover, this article examines three aspects, a brief conversation about the classification of integral equations, some important instruments as well as a real-world problem related with population growth is considered. Population growth increases due to the increase in the number of humans. For better understanding of the growth meaning, we can relate it with renewal and industrialization, more precisely, when food, water, energy and medical care become more available. As an expectation for the growth in population, however, due to the Covid-19 pandemic, millions of people worldwide have died since 2020. It has increased to 8.5 billion by 2030 and the global population will reach 9.7 billion in 2050, whereas 10.9 billion in 2100 [11]. In order to address population growth equation, three common fuzzy transforms are used for estimating its exact solution.

Simulation of a Marine Ecosystem Using Swarm Intelligence

The coexistence of multiple species and their interactions have been a topic of interest for hundreds of years. Humans have been taking advantage of species to maximize surplus or control our environment. Nowadays with the growing complexity of our effect on the Earth, it has become more difficult than ever to understand and plan how we are to use our environment. In this paper we attempt to simulate a marine ecosystem and investigate the factors Based on the boids algorithm and the population growth equation that are important for a sustainable system. This simulation will help us understand how marine populations grow and reach an equilibrium, how sudden spurts in one species affect the others and how even in the most devastating scenarios nature exhibits its incredible ability to replenish itself. We observed some interesting emergent behaviors and how the two species interacted with each other. Then with the use of such simulations, we plotted a population graph by collecting data of the number of sharks and fishes for each frame, and an equilibrium was created. As long as the predator relies on the prey, this equilibrium may have its ups and downs but will always stay in a stable range. This however becomes a different story when humans interfere with the food chain.

Chaos, percolation and the coronavirus spread: a two-step model.

We discuss a two-step model for the rise and decay of a new coronavirus (Severe Acute Respiratory Syndrome-CoV-2) first reported in December 2019, COVID-19. The first stage is well described by the same equation for turbulent flows, population growth and chaotic maps: a small number of infected, d_0, grows exponentially to a saturation value, d_infty . The typical growth time (aggressive spreading of the virus) is given by tau =frac{1}{lambda } where lambda is the Lyapunov exponent. After a time t_{mathrm{crit}} determined by social distancing and/or other measures, the spread decreases exponentially analogous to nuclear decays and non-chaotic maps. Some countries, like China, S. Korea and Italy, are in this second stage while others including the USA are near the end of the growth stage. The model predicted 15,000 (±2250) casualties for the Lombardy region (Italy) at the end of the spreading around May 10, 2020. Without the quarantine, the casualties would have been more than 50,000, one hundred days after the start of the pandemic. The data from the 50 US states are of very poor quality because of an extremely late and confused response to the pandemic, resulting unfortunately in a large number of casualties, more than 70,000 on May 6, 2020, and more than 170,000 on August 21, 2020. S. Korea, notwithstanding the high population density (511/hbox {km}^2) and the closeness to China, responded best to the pandemic with 255 deceased as of May 6, 2020, and 301 on August 21, 2020.

Intrashell Variability of Trace Elements in Benthic Foraminifera Grown Under High CO2 Levels

Two species of Amphistegina were cultured under four variable DIC concentrations (2340–2570 μM). The variability of trace elements within the foraminiferal shells was measured in the knob area of three individuals for each DIC treatment using LA-ICPMS. In individuals that showed significant growth (identified via 135Ba-enriched seawater), B, Na, and Sr showed an increase with DIC, while K and Mg were slightly lower or unchanged. Sharp transition zones between natural 135Ba and the ∼10-fold increased 135Ba in the shells represent one quarter of a new additional chamber, which occurs roughly once a week. The shape of the transition zone is best described by a logistic equation for population growth. We propose that this reflects the dynamics of seawater vacuoles that serve the biomineralization process and provide Ca and DIC for calcification of Amphistegina as described in previous publications (e.g., Bentov et al., 2009). LA-ICPMS profiles in the central knob (∼70 μm depth) also revealed previously described cyclical changes in concentration of Mg, each apparently representing a growth of a new chamber. Additional elements such as K, Na and U showed similar cycles with the same frequency and phase as the Mg cycles. Sr showed variability with similar frequency but not in-phase to those of the Mg. These multi-element cycles were found both in the newly grown calcite (elevated-135Ba) and in the natural skeleton regardless of the DIC treatments. These high Mg and multi-element cycles seem to be an essential part of the calcification process. They may originate from the interaction with the organic matrix resulting in elevated Mg and other elements in primary calcite while secondary calcite of the lamination process shows lower concentrations. It is also possible that primary calcite is enriched in trace elements if an Amorphous CaCO3 (ACC) or vaterite precursors are involved. In addition, Rayleigh fractionation from a semi-closed reservoir, the presence of high Mg in the lattice or any combination of the previous causes may explain the trace elements enrichment. Changes in the DIC did not affect the pattern of elemental cycles in these foraminifera, suggesting that this variability is inherent to the biomineralization process.

Analog simulator of integro-differential equations with classical memristors

An analog computer makes use of continuously changeable quantities of a system, such as its electrical, mechanical, or hydraulic properties, to solve a given problem. While these devices are usually computationally more powerful than their digital counterparts, they suffer from analog noise which does not allow for error control. We will focus on analog computers based on active electrical networks comprised of resistors, capacitors, and operational amplifiers which are capable of simulating any linear ordinary differential equation. However, the class of nonlinear dynamics they can solve is limited. In this work, by adding memristors to the electrical network, we show that the analog computer can simulate a large variety of linear and nonlinear integro-differential equations by carefully choosing the conductance and the dynamics of the memristor state variable. We study the performance of these analog computers by simulating integro-differential models related to fluid dynamics, nonlinear Volterra equations for population growth, and quantum models describing non-Markovian memory effects, among others. Finally, we perform stability tests by considering imperfect analog components, obtaining robust solutions with up to 13% relative error for relevant timescales.

Estimation of the Solution of the Kolmogorov-Fisher Type Biological Population Task by Taking Into Account the Reaction-Diffusion

The spatial-temporal dynamics of the population is one of the most interesting aspects and problems for environmental modeling. In this article, we will consider some mathematical models based on one-dimensional reaction-diffusion-advection equations for population growth in a heterogeneous habitat. Considering a number of models of increasing complexity, we investigate often the opposite roles of advection and diffusion for the conservation of the population. Whenever possible, we demonstrate basic mathematical methods and provide critical conditions that ensure the survival of the population, in simple systems and in more complex resourceconsumer models.

Breakthrough curve analysis by simplistic models of fixed bed adsorption: In defense of the century-old Bohart-Adams model

In the water and wastewater treatment field several simplistic models of packed bed dynamics such as the Bohart-Adams, Thomas, and Yoon-Nelson models are frequently used by investigators to fit adsorption breakthrough data. The century-old Bohart-Adams model is arguably the best known one which also serves as the foundation of the bed depth-service time equation. In recent years a substantial body of literature on the subject of fixed bed modeling has however claimed that it is inferior to other models. The present paper shows that such claims are incorrect and misleading because of biased comparisons in which the fitting ability of an oversimplified version of the Bohart-Adams model was compared with those of the Thomas and Yoon-Nelson models. The oversimplified Bohart-Adams equation is in effect an exponential function which predicts an exponentially increasing breakthrough value with time. As such, it is unable to fit typical breakthrough curves which are S-shaped or sigmoidal. It can be shown that a proper version of the Bohart-Adams model gives fit quality similar to those of the Thomas and Yoon-Nelson models. This is not unexpected since the three simplistic fixed bed models can be expressed in terms of the logistic equation of population growth; that is, mathematically they are equivalent.

BioGD: Bio-inspired robust gradient descent.

Recent research in machine learning pointed to the core problem of state-of-the-art models which impedes their widespread adoption in different domains. The models’ inability to differentiate between noise and subtle, yet significant variation in data leads to their vulnerability to adversarial perturbations that cause wrong predictions with high confidence. The study is aimed at identifying whether the algorithms inspired by biological evolution may achieve better results in cases where brittle robustness properties are highly sensitive to the slight noise. To answer this question, we introduce the new robust gradient descent inspired by the stability and adaptability of biological systems to unknown and changing environments. The proposed optimization technique involves an open-ended adaptation process with regard to two hyperparameters inherited from the generalized Verhulst population growth equation. The hyperparameters increase robustness to adversarial noise by penalizing the degree to which hardly visible changes in gradients impact prediction. The empirical evidence on synthetic and experimental datasets confirmed the viability of the bio-inspired gradient descent and suggested promising directions for future research. The code used for computational experiments is provided in a repository at https://github.com/yukinoi/bio_gradient_descent.

A reaction-diffusion population growth equation with multiple pulse perturbations

During the pest growth generations there exist a number of internal and external perturbations including birth pulses and pulse pesticide applications, which result in a complex growth pattern within each pest growth generation due to the variation of perturbation timings. In order to show how multiple pulse perturbations affect the dynamics of the pest population, we have extended the classical Fisher reaction-diffusion equation by involving instant birth and control perturbations. The main results indicate that the pulse perturbations have a remarkable influence on the existence and stability of a spatially homogeneous periodic solution. In particular, the birth rate and killing efficacy can affect the traveling wave and its spreading speed, while the timing of pesticide application does not make any effects on those threshold conditions. However, numerical investigations reveal that the timing of pesticide applications can significantly affect the spatial distributions of the pest population. Within each generation too early or too late pesticide application could result in a faster growth and wider spread of the pest, and the optimal timing of pesticide application is the middle of each generation provided that the efficacy of the pesticide is relatively low. Bifurcation analyses reveal that even generations and odd generations display a different spatial pattern, which depict some important issues related to the pest control and pest management, for example the different control strategies should be adopted for even and odd generations. To address how the spatial heterogeneity affects on the cost of the pest control, we further formulate the cost function to investigate the optimal pest control strategy, and the optimal spatial dependent killing rates which minimize the cost have been obtained numerically.

Estimating the sustainability of towed fishing‐gear impacts on seabed habitats: a simple quantitative risk assessment method applicable to data‐limited fisheries

Summary Impacts of bottom fishing, particularly trawling and dredging, on seabed (benthic) habitats are commonly perceived to pose serious environmental risks. Quantitative ecological risk assessment can be used to evaluate actual risks and to help guide the choice of management measures needed to meet sustainability objectives. We develop and apply a quantitative method for assessing the risks to benthic habitats by towed bottom‐fishing gears. The method is based on a simple equation for relative benthic status (RBS), derived by solving the logistic population growth equation for the equilibrium state. Estimating RBS requires only maps of fishing intensity and habitat type – and parameters for impact and recovery rates, which may be taken from meta‐analyses of multiple experimental studies of towed‐gear impacts. The aggregate status of habitats in an assessed region is indicated by the distribution of RBS values for the region. The application of RBS is illustrated for a tropical shrimp‐trawl fishery. The status of trawled habitats and their RBS value depend on impact rate (depletion per trawl), recovery rate and exposure to trawling. In the shrimp‐trawl fishery region, gravel habitat was most sensitive, and though less exposed than sand or muddy‐sand, was most affected overall (regional RBS = 91% relative to un‐trawled RBS = 100%). Muddy‐sand was less sensitive, and though relatively most exposed, was less affected overall (RBS = 95%). Sand was most heavily trawled but least sensitive and least affected overall (RBS = 98%). Region‐wide, >94% of habitat area had >80% RBS because most trawling and impacts were confined to small areas. RBS was also applied to the region's benthic invertebrate communities with similar results. Conclusions. Unlike qualitative or categorical trait‐based risk assessments, the RBS method provides a quantitative estimate of status relative to an unimpacted baseline, with minimal requirements for input data. It could be applied to bottom‐contact fisheries world‐wide, including situations where detailed data on characteristics of seabed habitats, or the abundance of seabed fauna are not available. The approach supports assessment against sustainability criteria and evaluation of alternative management strategies (e.g. closed areas, effort management, gear modifications).

Exploring Phytoplankton Population Investigation Growth to Enhance Quantitative Literacy

Quantitative literacy is essential to biological literacy (and is one of the core concepts in Vision and Change in Undergraduate Biology Education: A Call to Action;AAAS 2009). Building quantitative literacy is a challenging endeavor for biology instructors. Integrating mathematical skills into biological investigations can help build quantitative literacy. In our plankton population laboratory sequence, students test hypotheses about the influence of abiotic factors on phytoplankton populations by sampling experimental and control flasks over multiple weeks. Students track and predict changes in planktonic populations by incorporating weekly sample estimates into population growth equations. We have refined the laboratory protocols on the basis of student commentary and instructor observations. Students have reviewed the lab positively, and approximately one-quarter of them reported building their math skills by participating in the lab.

In vitro와 in vivo에서의 온도에 따른 스쿠티카충 성장의 수리 모델

실험실에서 온도 별로 스쿠티카충을 배양한 데이터(Table 1)를 사용 스쿠티카충 개체군의 변화율(1.2)과 성장 방정식(1.3)을 in vitro 조건에서 구했다. 이 모델에서는 스쿠티카충의 증가모델 식은 로지스틱 증가함수로 두 가지 온도에서의 스쿠티카충의 증가율 매개변수를 구하였고, 기생충으로 인한 숙주인 넙치에 치명적일 수 있는 기생충의 임계밀도를 구하였다. 또 다른 데이터는 같은 온도에서 스쿠티카충 수를 다양하게 넙치의 복강에 주사하는 그룹과 동일한 스쿠티카충 수를 복강에 주사한 후 온도를 다르게 한 두 그룹의 스쿠티카충의 감염으로 폐사한 실험데이터(Table 2)에 근거하여 넙치의 감염에서 폐사가 되는 시스템 식(1.5)을 세웠다. 정리에서는 감염된 넙치와 폐사된 넙치의 미분방정식계는 감염넙치와 폐사 넙치와의 평형점을 찾고 그 평형점에 대한 안정성과 불안정성을 설명하였다. 각 온도에서의 매개변수를 사용하면 방정식 (1.6)은 시간에 대해 넙치의 누적 폐사량을 알 수 있는 방정식이다. 이 논문에서는 다양한 초기값 <TEX>$P_0$</TEX>값에 따른 넙치의 폐사를 중심으로 보았으므로 온도와 <TEX>$P_0$</TEX>가 동시에 넙치폐사에 어떠한 관계를 갖는 연구를 하기에는 제약을 갖고 있다. 이 연구의 제한점에서 언급했듯 실험을 여러 번 시행한 것이 아니기 때문에 단지 근사적인 값으로 이해를 해야 한다. 또한 넙치의 면역력을 알 수 있다면 기생충으로부터 저항력을 연구하는데 오차를 좀 더 줄인 근사치를 구할 수 있을 것을 기대할 수 있을 것이다. Population growth equation of scuticociliate Miamiensis avidus was obtained from the experimental results of in vitro culture condition to estimate the growth rate and carrying capacity from the growth equation. In addition, intraperitoneal infections into olive flounder Paralichthys olivaceus were carried out into 2 different conditions: different concentrations of M. avidus in same water temperature and same concentration of M. avidus in different water temperatures. Olive flounder mortality was threshold dependent with both the temperature and M. avidus density parameters. In this paper, we propose a mathematical model to study M. avidus growth in olive flounder based upon the interactions between parasite and host. The mathematical model was logistic growth differential equation (1.2). The parameters were found with Matlab program through the Levenberge-Marquardt method. In theorem, equilibrium values between the infected fish population and dead population could found. Our equilibrium points were a stable equilibrium and an unstable equilibrium. From the equation (1.6), it was possible to predict the amount of cumulative mortality of olive flounder along with the time after M. avidus infection.

Integration of logistic and kinetics equations of population growth

The procedure proposed by Sakanoue [Sakanoue, S., 2007. Extended logistic model for growth of single-species populations. Ecological Modelling 205, 159–168] is used to derive the kinetics equations of population growth. It is based on three assumptions: resource availability changes with population size; resource supply to population and population demand for resources are defined as functions of resource availability and population size; and resource availability and population size shift in the supply function attracted to the demand function. These assumptions are organized into an abstract equation, which can be transformed into the Verhulst logistic equation under certain supply and demand functions. On the other hand, by setting “per capita resource availability” as an independent variable, the abstract equation can also be transformed into some existing kinetics equations and new kinetics ones involving intraspecific interactions such as facilitation and interference. The procedure provides a unified means of deriving and relating the two types of population growth equation.

Accounting for wildlife life‐history strategies when modeling stochastic density‐dependent populations: A review

AbstractThis article briefly reviews and provides discussion on the evidence for, and nature of, density‐dependence patterns in r and K‐selected species. In this review, I discuss how life‐history strategies cause different nonlinear density‐dependence patterns and I provide a simple modeling recommendation to incorporate nonlinear density dependence in population growth equations. Second, I discuss the importance of incorporation of environmental stochasticity and local extinction associated with nonlinear density dependence associated with life‐history patterns through a novel modeling exercise. Last, I discuss the importance of considering how life‐history nonlinear density dependence could affect optimal harvest yields. Though these topics are extensive, this review should spur wildlife biologists and managers to consider more inclusive population models that incorporate life‐history strategies and stochasticity in their decision‐making processes. © 2012 The Wildlife Society.

An economic growth model with endogenous carrying capacity and demographic transition

In this paper, a mathematical model is set up to inquire population change under interaction between the economic growth and human population carrying capacity. By introducing the population growth equation with variable carrying capacity into the classical Solow model and combining the population growth equation, we obtain a two-dimensional dynamical system. It is proved that the dynamical system has a unique equilibrium and the solution of the dynamical system is asymptotically stable. By qualitative analysis, we obtain that the population growth rate increases from zero to a positive level firstly and then decreases to zero and per capita capital increases strictly along a normal economic growth path. Therefore, the model implies that the demographic transition appears under the interaction between economic growth and human population carrying capacity.

The effects of climate change on density‐dependent population dynamics of aquatic invertebrates

Global climate change has the potential to alter aquatic communities through changes in evapotranspiration and increased variability in precipitation. We used aquatic mesocosms to test the impacts of variable precipitation on population dynamics of common mosquito (Culicidae) and midge (Chironomidae) larvae that inhabit vernal pools. In a mixed deciduous forest in northern Vermont, USA, we orthogonally crossed seven levels of mean water level (increased rainfall) with seven levels of water level coefficient of variation (more variable rainfall) to simulate a broad array of climate change scenarios in 49 experimental mesocosms.The average abundance of Culicidae was highest at low water levels, whereas the average abundance of Chironomidae was highest at higher water levels and low variability in water level. Treatments and environmental and spatial covariates collectively explained 49% of the variance in mean abundance. For both taxa, we fit hierarchical Bayesian models to each 16‐week time series to estimate the parameters in a Gompertz logistic equation of population growth with density dependence. We found that Culicidae population growth rate increased with decreasing water levels and that 87% of the variance in Chironomidae density dependence could be explained by treatment. Collectively, these results suggest that climate change can alter abundances aquatic invertebrate taxa but not necessarily through the same mechanism on all populations. In the case of Culicidae the abundance is affected by changes in growth rate, and in Chironomidae by changes in the strength of density dependence.

Bifurcation analysis of the flour beetle population growth equations

In this article, we study a discrete delayed flour beetle population equation. Firstly, we study the existence of period-doubling bifurcation and Neimark–Sacker bifurcations for the system by analysing its characteristic equations. Secondly, we investigate the direction of the two bifurcations and the stability of the bifurcation periodic solutions by using normal form theory. Finally, some numerical simulations are carried out to support the analytical results.

Anisotropic dispersion, space competition and the slowdown of the Neolithic transition

The front speed of the Neolithic (farmer) spread in Europe decreased as it reached Northern latitudes, where the Mesolithic (hunter-gatherer) population density was higher. Here, we describe a reaction–diffusion model with (i) an anisotropic dispersion kernel depending on the Mesolithic population density gradient and (ii) a modified population growth equation. Both effects are related to the space available for the Neolithic population. The model is able to explain the slowdown of the Neolithic front as observed from archaeological data.

Early warning signals of extinction in deteriorating environments

During the decline to extinction, animal populations may present dynamical phenomena not exhibited by robust populations. Some of these phenomena, such as the scaling of demographic variance, are related to small size whereas others result from density-dependent nonlinearities. Although understanding the causes of population extinction has been a central problem in theoretical biology for decades, the ability to anticipate extinction has remained elusive. Here we argue that the causes of a population's decline are central to the predictability of its extinction. Specifically, environmental degradation may cause a tipping point in population dynamics, corresponding to a bifurcation in the underlying population growth equations, beyond which decline to extinction is almost certain. In such cases, imminent extinction will be signalled by critical slowing down (CSD). We conducted an experiment with replicate laboratory populations of Daphnia magna to test this hypothesis. We show that populations crossing a transcritical bifurcation, experimentally induced by the controlled decline in environmental conditions, show statistical signatures of CSD after the onset of environmental deterioration and before the critical transition. Populations in constant environments did not have these patterns. Four statistical indicators all showed evidence of the approaching bifurcation as early as 110 days (∼8 generations) before the transition occurred. Two composite indices improved predictability, and comparative analysis showed that early warning signals based solely on observations in deteriorating environments without reference populations for standardization were hampered by the presence of transient dynamics before the onset of deterioration, pointing to the importance of reliable baseline data before environmental deterioration begins. The universality of bifurcations in models of population dynamics suggests that this phenomenon should be general.

Kinetic growth models at different biological development levels

The quasi-chemical biological growth model suggested earlier was used to derive kinetic equations for population growth of various biological species. A cycle of sequential stages of population development can be represented in the form of quasi-chemical equations; the kinetic equations describing the development cycle allow the population size changes to be predicted quantitatively depending on the conditions. As distinct from the extrapolation-simulation model, the kinetic parameters in kinetic biological growth models are the experimental population growth characteristics. The models developed were verified and parametrically identified on the basis of the experimental data to show that they were in agreement with experiment to within measurement errors.