MODEL LOGISTIK FUZZY DENGAN ADANYA PEMANENAN PROPORSIONAL

Having much attention in the past few years, uncertainty (fuzzy, interval etc.,) play important role in mathematical biology. In this paper we study a proportional harvesting model in fuzzy environment. The model is considered in three different way: initial population density as a fuzzy number, intrinsic growth rate and proportional harvesting are fuzzy number and initial population density, intrinsic growth rate, proportional harvesting are fuzzy number. The solution procedure is done by the concept of fuzzy differential equation. This paper explores the stability analysis of a bio mathematical model in fuzzy environment. We have discussed about equilibrium points and their feasibility. The necessary numerical result is done and discuss briefly.

This paper presents the effects of two well known harvesting models namely quota harvesting and proportional harvesting on the logistic growth model under fuzziness. In many real life situations it is not possible to have a single growth rate of a species throughout the specified time span. Species are enhanced at different growth rates during their life span that arises due to nature of the soil, whether condition, rain fall, food supply etc. Similarly the total harvesting quantity for quota harvesting model and the proportion of the harvesting quantity for proportional harvesting quantity depend on the nature (need) of the harvester. So these are imprecise in nature, here we assume that the growth rate, constant harvesting quantity and proportion of harvesting quantity are uncertain in reality by considering imprecise by triangular fuzzy number. We find the equilibrium points, bifurcation value and discuss the stability of the equilibrium points of the models under fuzziness with different perspectives. Numerical examples are given to illustrate the harvesting models under fuzziness. Finally graphical illustrations are given to study the effects of harvesting on the logistic growth model under fuzziness.

Dynamical behaviour of discrete logistic equation with Allee effect in an uncertain environment

Richards model, Gompertz model, and logistic model are widely used to describe growth model of a population. The Richards growth model is a modification of the logistic growth model. In this paper, we present a new modified logistic growth model. The proposed model was derived from a modification of the classical logistic differential equation. From the solution of the differential equation, we present a new mathematical growth model so called a WEP-modified logistic growth model for describing growth function of a living organism. We also extend the proposed model into couple WEP-modified logistic growth model. We further simulated and verified the proposed model by using chicken weight data cited from the literature. It was found that the proposed model gave more accurate predicted results compared to Richard, Gompertz, and logistic model. Therefore the proposed model could be used as an alternative model to describe individual growth.

The present study was conducted to estimate and compare the three types of growth models in Hanwoo steer (Bos aurus coreanae). The Gompertz, Von Bertalanffy, and Logistic nonlinear models were used. A total of 2,239 Hanwoo steers (Bos taurus coreanae) from 6 months to 24 months old (2003 to 2014) and 8,916 growth data from the Hanwoo improvement Center were used to estimate the growth model which included three parameters. These parameters were A, mature body weight; b, growth ratio; and k, intrinsic growth rate. Regression equations using the Gompertz, Von Bertalanffy, and Logistic models were calculated as respectively. The mean square errors (MSEs) for each model were 1945.9, 1958.7, and 1935.0, respectively. The equation using the Logistic model showed the lowest value among three models. The estimated birth weights from the Gompertz, Von Bertalanffy, and Logistic models were 50.35 kg, 36.94 kg, and 74.13 kg, respectively. Furthermore, the estimated mature weights from the Gompertz, Von Bertalanffy, and Logistic models were 919.0 kg, 1043.3 kg, and 770.0 kg, respectively. In addition, the estimated age and body weight at inflection from the Gompertz, Von Bertalanffy, and Logistic models were 349.0 days and 338.1 kg, 317.9 days and 308.2 kg, and 397.8 days and 385.0 kg, respectively. Based on the results, we concluded that the regression equation using the Logistic model was the most appropriate among the growth models for measuring data. However, further studies would be needed in order to obtain more accurate parameters using a much wider period of data from birth to shipping age.

The classical population growth models include the Malthus population growth model and the logistic population growth model, each of which has its advantages and disadvantages. To address the disadvantages of the two models, this paper establishes a grey logistic population growth prediction model, based on the modeling mechanism of the grey prediction model and the characteristics of the logistic model, which uses the least-squares method to estimate the maximum population capacity. In accordance with the data characteristics of population growth, the weakening buffer operator is used to establish the weakening buffer operator grey logistic population growth prediction model, which improves its accuracy, thus improving the classic population prediction model. Four actual case datasets are used simultaneously, and the two classical grey prediction models are compared. The results of the six evaluation indicators show that the effects of the new model demonstrate obvious advantages. Finally, the new model is applied to the population forecast of Chongqing, China. The prediction results suggest that the population may reach a peak in 2020 and decline in the future. This finding is consistent with the logistic population growth model.

Everyone knows the logistic model of population growth. Populations cannot grow indefinitely; hence the per capita growth rate must decrease as the population density N increases; the simplest decreasing function is linear; and so we have d N dt = [r(1 − N/K )]N , where r is the intrinsic growth rate and K is called the carrying capacity of the environment. Conscientious textbooks will add in parentheses, “(assuming r > 0)”. Indeed, with r < 0, the equation predicts unbounded growth, a biological nonsense, for populations whose initial size exceeds K . But an experimentalist is free to choose any positive initial density, and r < 0 is certainly not nonsense: r < 0 holds when the death rate exceeds the birth rate even at low population densities, as for an unfit mutant or a species outside its region of viability. This shows that we cannot meaningfully vary r and K independently. But how they should be connected is unclear: The information is simply not in the equation. A population is an ensemble of individuals, and its behaviour is ultimately a consequence of the behaviour of the individuals. In order to predict population behaviour, we should first describe the behaviour of the individuals and then use it to derive a model of the population. When the logistic model is derived in this way, it is clear that K is not an externally fixed parameter (the “carrying capacity of the environment”).

The phenomenon of the spread of infectious disease can be formed as an epidemic model. Simplest epidemic model is SI model which can be extended to SIR and SIS model. If recover person will not be susceptible to same disease until the immunity dies out, then the model will be SIRS model. If all person in SIRS model are in population which have restrictiveness of carrying capacity, then it will be formed SIRS model with logistic growth. This model can be presented mathematically using a system of differential equations. Based on SIRS model with logistic growth, it is obtained three disease-free equilibrium points and one endemic equilibrium point. Two disease-free equilibrium points are not stable, while one more point is asymptotically stable if modified reproduction ratio number more than one and the ratio between intrinsic growth rate with death rate which is caused of disease less than proportion of number of infected person with total population. Based on modified reproduction ratio number, loss of disease from population is affected by the interaction rate between susceptible person with infected person, recovery rate of infected person, death rate which is caused of disease, and birth rate. An endemic equilibrium point is asymptotically stable if basic reproduction ratio number more than one. Based on basic reproduction ratio number, epidemic case in population are affected by several things, they are the interaction rate between susceptible person with infected person, recovery rate of infected person, death rate which is caused of disease, birth rate, the factor which influence birth rate decrease, and intrinsic growth rate.

Integrating the standardization of abundance indices into stock assessment models to examine the population dynamics of small yellow croaker, Larimichthys polyactis, was tested through a fisheries mixture matrix constructed with multiple data types. A precautionary approach to fishery control rules was adopted based on the logistic and Fox surplus production models, incorporating data from fishery-independent surveys, fisherydependent catch-per-unit-of-effort(CPUE), and regional harvests. A risk-averse control rule, derived from model parameters and associated uncertainty, was developed to manage fisheries for maximum sustainable yield(MSY)and rapid rebuilding of overfished stocks. The proposed control rule consists of relative biomass and relative fishing mortality rate in a deterministic environment and conservative harvest in a fluctuating environment. The results of the Fox model explained 68% of the variance observed for the stock abundance, while the logistic model explained 57%. The parameter estimates were different and the Fox model predicted a much larger decrease in population abundance at the MSY, intrinsic growth rates(r), and initial exploited levels. We compared the fishing mortality/current stock biomass from 1998 to 2006 with the fishing- and stock-related reference points, respectively. The results in a determined environment revealed that small yellow croaker stock in the East China Sea was overfished in most years, while the population was not always overfished during the entire period, although its biomass has been decreasing since 1999. However, both the Fox and logistic surplus production models indicate that the small yellow croaker fishery has been consistently over harvested in the fluctuating environment. Harvesting at a conservative level with either the Fox or logistic model could increase small yellow croaker abundance substantially with little decrease in harvest. At a conservative harvest level, there is a 24.7% increase in biomass with a 6.1% decrease in yield with the logistic model and a 44.5% increase in biomass with a 9.4% decrease in yield with the Fox model. The MSY assessment results from the Fox surplus production model was more conservative than that of logistic model, which is concordant with precautionary fisheries management strategies.

This paper studies the effect of time delay and proportional harvesting on stability of mutualism model as well as to establish conditions for sustainable harvesting so that the unique positive equilibrium point is stable. First the stability of the model with harvesting is analyzed. The model is then improved by considering a time delay in the mechanism of the growth rate of the population. The analysis shows that when there is no delay, varying the harvesting rate does not affect the stability of the model if the value of harvesting rate is under control while time delay can cause a stable equilibrium point to become unstable.

Most harvest theory is based on an assumption of a constant or stochastic environment, yet most populations experience some form of environmental seasonality. Assuming that a population follows logistic growth we investigate harvesting in seasonal environments, focusing on maximum annual yield (M.A.Y.) and population persistence under five commonly used harvest strategies. We show that the optimal strategy depends dramatically on the intrinsic growth rate of population and the magnitude of seasonality. The ordered effectiveness of these alternative harvest strategies is given for different combinations of intrinsic growth rate and seasonality. Also, for piecewise continuous-time harvest strategies (i.e., open/closed harvest, and pulse harvest) harvest timing is of crucial importance to annual yield. Optimal timing for harvests coincides with maximal rate of decline in the seasonally fluctuating carrying capacity. For large intrinsic growth rate and small environmental variability several strategies (i.e., constant exploitation rate, linear exploitation rate, and time-dependent harvest) are so effective that M.A.Y. is very close to maximum sustainable yield (M.S.Y.). M.A.Y. of pulse harvest can be even larger than M.S.Y. because in seasonal environments population size varies substantially during the course of the year and how it varies relative to the carrying capacity is what determines the value relative to optimal harvest rate. However, for populations with small intrinsic growth rate but subject to large seasonality none of these strategies is particularly effective with M.A.Y. much lower than M.S.Y. Finding an optimal harvest strategy for this case and to explore harvesting in populations that follow other growth models (e.g., involving predation or age structure) will be an interesting but challenging problem.

In the recent years much importance has been laid on the role of uncertainty (fuzzy, interval etc.) in mathematical biology. In this paper we tried to study on the quota harvesting model in fuzzy environment. This model is considered in three different ways viz. (1) Initial condition (population density) is a fuzzy number, (2) coefficients of quota harvesting model (intrinsic growth rate and quota harvesting rate) are fuzzy number and (3) both initial condition and coefficients are fuzzy number. We discuss all these fuzzy cases individually. The solution procedure is done by using the concept of fuzzy differential equation approach. We have discussed the equilibrium points and their feasibility in all the three cases. This paper explores the stability analysis of the quota harvesting model at the equilibrium points in fuzzy environment. In order to examine the stability systematically in different fuzzy cases, we have used numerical simulations and discussed them briefly.

Plants are known to maintain fitness despite herbivore attack by a variety of damage-induced mechanisms. These mechanisms are said to confer tolerance, which can be measured as the slope of fitness over the proportion of plant biomass removed by herbivore damage. It was recently supposed by Stowe et al. (2000) that another plant property, general vigor, has little effect on tolerance. We developed simple models of annual monocarpic plants to determine if a genetic change in components of growth vigor will also change the fitness reaction to damage. We examined the impact of intrinsic growth rate on the tolerance reaction norm slope assuming plants grow geometrically, i.e., without self-limitation. In this case an increase in intrinsic growth rate decreases tolerance (the reaction norm slope becomes more negative). A logistic growth model was used to examine the impact of self-limiting growth on the relationship between intrinsic growth rate and the tolerance reaction norm slope. With self-limitation, the relationship is sensitive to the timing of attack. When attack is early and there is time for regrowth, increasing growth rate increases tolerance (slope becomes less negative). The time limitations imposed by late attack prevent appreciable regrowth and induce a negative relationship between growth rate and tolerance. In neither of these simple cases will the correlation between vigor and tolerance constrain selection on either trait. However, a positive correlation between growth rate and self-limitation will favor fast growth/strong self-limitation in a high-damage environment, but slow growth/weak self-limitation in a low-damage environment. Thus, fundamental growth rules that determine vigor have constitutive effects on tolerance. The net costs and benefits of damage-induced tolerance mechanisms will thus be influenced by the background imposed by fundamental growth rules.

Models of population dynamics are substantially non-Markovian in nature and exhibit behavior for memory effects. This research study investigates the logistic growth model from population dynamics under both classical and non-classical (fractional) differential operators using actual statistical data. For the non-classical differential operator, the operator called conformable fractional derivative having order β in the sense of Liouville-Caputo (LC) of order α is employed while taking care of its dimensional consistency. Working parameters including conformable fractional derivative order β, LC having order α, growth rate ζ, and carrying capacity Φ have been optimized, and the results are compared with those of classical one. Error rate suggests the superiority of the fractional conformable logistic model whose solutions are investigated for uniqueness via fixed point theory. Numerical simulations using Adams' iterative technique recently proposed for the conformable fractional derivative are illustrated in figures for a thorough understanding of the model's behavior, along with a statistical summary for the operators. An attractive chaotic attractor is observed in the fractional logistic model when ζ ≥ 4.023. Cumulative effects of growth rate and carrying capacity on the population's change in the logistic model are also investigated with 3D graphics.

The purpose of this study focused on modeling the population of Ethiopia using different models and estimating the models parameters via least square method. The models, that were applied for the population growth, were Malthus growth model, Logistic growth model and General growth model. To identify the models which performed effectively in prediction of the actual population, the measure accuracy has been used, such that the models satisfying the criteria of the measure of accuracy is the best statistical model. The results of the analysis were presented using tables and graphical form which are very good to perform comparison for the effectiveness of the models. In this study, MAPE, RSE , MAD and R2 which are considered to measure the accuracy of the models. Malthus growth model, Logistic growth model and General growth model used the population of Ethiopia from 1980 to 2020 inclusive, the data was obtained from international data base(IDB). R studio 3.6.3 were used to estimate the models parameters using simple codes. The study proposed to project the population of Ethiopia via General growth model which performed best in measure of accuracies that makes it effective and efficient as compare with the other models. The model had the smallest RSE ( 492,155 ), MAPE (0.75%) and MAD ( 379,942 ) as well as the highest R2(99.97%) relative to the other models. Keywords: Logistic model, parameter estimation, Malthus model, General model, selection methods DOI: 10.7176/JNSR/12-1-05 Publication date: January 31 st 2021