Analysis of Stability of Some Population Models with Harvesting

Applied mathematics, which means application of mathematics to problems, is a wonderful and exciting subject. It is the essence of the theoretical approach to science and engineering. It could refer to the use of mathematics in many varied areas. Mathematical model is applied to predict the behaviour of the system. This behaviour is then interpreted in terms of the word model so that we know the behaviour of the real situation. We can apply mathematical languages to transform ecology's phenomena into mathematical model, including changes of popUlations and how the populations of one system can affect the population of another. The model is expected to give us more information about the real situation and as a tool to make a decision. Some models that constitute autonomous differential equations are presented; Malthusian and logistic model for single population; two independent populations, competing model, and prey-predator model for two populations; and extension of prey-predator model involving three populations. In this thesis we will study the effect of harvesting on models. The models are based on Lotka-Volterra model. All models involve harvesting problem and some stable equilibrium points related to maximum profit or maximum sustainable yield problem. The objectives of this thesis are to analyse, to investigate the stability of equilibrium point of the models and to control the exploitation efforts such that the population will not vanish forever although being exploited. The methods used are linearization method, eigenvalues method, qualitative stability test and Hurwitz stability test. Some assumptions are made to avoid complexity. Maple V software release 4 is used to determine the equilibrium points of the model and also to plot the trajectories and draw the surface. The single population model is solved analytically.We found that in single population model, the existence of population depends on the initial population and harvesting rate. In model that involves two and three populations, the populations can live in coexistence although harvesting is applied. The level of harvesting, however, must be strictly controlled

Stability Analysis of Some Population Models With Time Delay and Harvesting

This research presents the development and extension of some models for the growth rates of population. The existing models, i.e., logistic population model for single population, predator – prey model, Wangersky – Cunningham model, competing model and symbiosis model for two interaction populations, are extended by considering time delay, harvesting function and time delay in harvesting term in the models to get some new population models. The time delay is considered in the model to make the model more accurate because the growth rate of population does not only depend on the present size of population but also depends on past information. The current size of the population does not immediately change the growth rate of the population, but there is a time delay. The population as a valuable stock, for example fish population, is then harvested. The considered harvesting functions in the new population models are constant effort and constant quota of harvesting. The new models are then analyzed to determine the stability of their equilibrium points. Before determining the stability of the equilibrium points, we provide the necessary and sufficient conditions for the existence of the equilibrium points. Since we consider population model, we just investigate the nonnegative equilibrium points. For some models, we determine only the sufficient conditions for the existence of the positive equilibrium points. The value of time delay, level of harvesting, initial size of populations, and parameters of the models need to be controlled so that the populations will not be extinct for a long time and also the populations give maximum profit. The methods used to study the stability of the equilibrium point are linearization model around the equilibrium point, eigenvalues method, phase plane analysis, and plotting trajectories around the equilibrium point. In order to determine the stability of the equilibrium point, we inspect the sign of real parts of the eigenvalues. The graphs of the trajectories are plotted to visualize the behavior of the trajectories. For the models with constant effort of harvesting, we determine the critical value of the effort that maximizes the profit and does not affect the stability of the equilibrium point. Some new theorems are constructed and proved to determine the time delay margin, stability switches and stability intervals. We find that there exists a certain condition so that the positive equilibrium point of the models becomes stable. From the analysis we find that the time delay can induce instability, stability switches and bifurcations in all the models except for the symbiosis model with time delay in harvesting term. The analysis also shows that for the models with constant effort of harvesting, there exists a critical value for the effort of harvesting that maximizes the profit function and maintains the stability of the equilibrium point. When we control the values of the parameters, level of harvesting, and time delay, the positive equilibrium point can be found and possibly stable. The existence of the populations also depends on the initial value of the population since we just consider local stability. For the models without time delay and harvesting, we find the global stability of the positive equilibrium point. For the models with a time delay, there exists either a time delay margin or some stability switches so that the positive equilibrium point remains stable on the stability interval. The maximum profit can be found without affecting the stability of the equilibrium point when the values of parameters and the level of constant efforts of harvesting are strictly controlled

Model Perubahan Sub Populasi (Menikah Dan Tidak Menikah) Dalam Populasi Manusia

Pada tulisan ini dibahas suatu model perubahan populasi manusia. Populasi manusia dibagi dalam tiga sub populasi yaitu; populasi pria yang tidak menikah, populasi wanita yang tidak menikah dan populasi yang menikah. Perubahan masing-masing sub populasi dipengaruhi oleh jumlah sub populasi lainnya. Beberapa asumsi dibuat untuk keperluan pemodelan. Model perubahan sub populasi itu dikonstruksi dan dinyatakan dalam bentuk sistem persamaan diferensial orde satu. Syarat kewujudan suatu titik keseimbangan positif diberikan. Kestabilan titik keseimbangan positif dari model dianalisis dengan melinearkan model di sekitar titik keseimbangan dan dengan memeriksa nilai eigen dari persamaan karakteristik. Analisis kestabilan menunjukkan bahwa mungkin wujud suatu titik keseimbangan positif yang bermakna bahwa masing-masing jumlah sub populasi akan menuju ke suatu nilai positif tertentu

Analisis Kestabilan Model Logistik Satu Populasi Dengan Tundaan Waktu

Dalam tulisan ini dibahas model logistik satu populasi dengan tundaan waktu. Tundaan waktu dipertimbangkan dalam model karena laju perubahan suatu populasi tidak hanya bergantung pada jumlah populasi pada saat sekarang tetapi juga bergantung pada jumlah populasi pada waktu lampau. Pada tulisan ini pengaruh tundaan waktu terhadap kestabilan titik keseimbangan model dianalisis dengan menggunakan pendekatan deret Taylor di sekitar titik keseimbangan. Dengan pendekatan itu diperoleh suatu model tanpa bentuk jumlah populasi yang bergantung pada tundaan waktu tetapi model tersebut tetap bergantung pada tundaan waktu. Penyelesaian analitik untuk model tersebut diberikan. Dari penyelesaian analitik itu diketahui bahwa titik keseimbangan positif stabil untuk suatu kondisi tundaan waktu dan jumlah awal populasi tertentu.

Stability Analysis of Wangersky-Cunningham Model with Constant Effort of Harvesting

In this paper we consider another predator-prey model with time delay which is called the Wangersky-Cunningham model. In this model, the rate of change of the predator population depends on the numbers of prey and predator present at some previous time. The model is then improved by considering a constant effort of harvesting into the growth rate of the prey and predator populations. The method use in this analysis is linearization the model around the equilibrium point and then inspecting the eigenvalues to determine the stability. We found that there exists a positive equilibrium point for the model with and without harvesting. The time delay can induce instability and Hopf bifurcation can also occur. Some plots of trajectories of the prey and predator populations are also given

Analisis Kestabilan Dan Keuntungan Maksimal Pada Model Pertumbuhan Populasi Mangsa-Pemangsa Dengan Tahapan Struktur

Artikel ini membahas model pertumbuhan populasi mangsa-pemangsa. Model menyatakan laju pertumbuhan tiga populasi, yaitu populasi mangsa, populasi pemangsa belum dewasa, dan populasi pemangsa sudah dewasa. Perubahan ukuran populasi mangsa dipengaruhi oleh pertumbuhan intrinsik dan interaksinya dengan populasi pemangsa dewasa. Perubahan ukuran populasi pemangsa dewasa dipengaruhi oleh interaksinya dengan populasi mangsa, kematian alamiah, dan perpindahan populasi pemangsa belum dewasa menjadi pemangsa dewasa. Perubahan populasi pemangsa belum dewasa dipengaruhi oleh jumlah kelahiran dari populasi pemangsa dewasa, kematian alamiah, dan perubahan populasi pemangsa menjadi pemangsa dewasa. Dinamika ketiga populasi tersebut dinyatakan dalam bentuk sistem persamaan differensial orde satu yang menyatakan perubahan ukuran populasi terhadap waktu. Dengan menganggap bahwa populasi yang ditinjau bernilai ekonomi, maka ketiga populasi tersebut dieksploitasi. Selanjutnya model melibatkan fungsi pemanenan pada perubahan ketiga populasi. Kewujudan titik ekuilibrium dari model beserta kestabilannya dianalisis dengan menggunakan metode linearisasi dan uji kestabilan Routh-Hurwitz. Hal ini dilakukan untuk menjamin ketiga populasi tidak akan punah dalam jangka waktu yang panjang. Selain itu, akan dianalisis usaha pemanenan optimal yang digunakan dalam mengeksploitasi populasi sehingga diperoleh keuntungan maksimal dan ketiga populasi tetap akan lestari dalam jangka waktu yang panjang. Beberapa kasus dianalisis yang disertai dengan simulasi numerik untuk mengetahui kestabilan titik ekuilibrium dan keuntungan maksimal. Hasil analisis menunjukkan bahwa kewujudan dan kestabilan titik ekuilibrium interior pada model ditentukan oleh nilai-nilai paramater model dan usaha pemanenan. Ketiga populasi dapat tetap lestari meskipun dieksploitasi dengan usaha pemanenan konstan dan sekaligus memberikan keuntungan maksimal

Model Matematika Kemotaksis dalam Penyakit Alzheimer

Alzheimer merupakan penyakit demensia yang disebabkan oleh Senile plaques yang merupakan akumulasi protein ??-amyloid sebagai akibat produksi yang terlalu tinggi. Proses ini juga menyebabkan pengaktifan sel microglia atau inflamatory, akibatnya microglia akan bergerak menuju sumber ??-amyloid, peristiwa ini disebut kemotaksis. Peristiwa kemotaksis telah dirumuskan ke dalam suatu model matematika oleh Keller-Segel pada 1970 yang kemudian dikembangkan pada kasus penyakit Alzheimer oleh M. Luca pada 2001. Pada penelitian ini model tersebut dikembangkan dan solusi numeriknya dianalisis dengan metode beda hingga. Hasil yang diperoleh menunjukkan bahwa konsentrasi microglia dan ??-amyloid akan homogen untuk waktu yang lam

Model Dengan Tundaan Waktu

Pada tulisan ini dibahas model matematika yang melibatkan tundaan waktu, termasuk bagaimana munculnya tundaan waktu dalam suatu model dan urgensinya dilibatkan dalam model. Selanjutnya diberikan beberapa model yang melibatkan tundaan waktu. Beberapa metode yang biasa digunakan untuk menyelesaikan model dengan tundaan waktu, dan metode-metode yang digunakan dalam menganalisis kestabilan titik keseimbangan suatu model dengan tundaan waktu juga diberika

Stability Analysis of Prey-Predator Population Model with Harvesting on The Predator Population

In this paper we present a deterministic and continuous model for one prey–one predator population model based on Lotka-Volterra model. The predator population is subjected to both constant effort and constant quota of harvesting. We study analytically the sufficient conditions of harvesting to ensure the stability of the equilibrium point. The method used to analyze the stability of the equilibrium point is linearization and Hurwitz stability test. The results show that the equilibrium point which occurs in positive quadrant is stable although the predator population is subjected to harvesting. This means that the prey and predator populations can live in coexistence although the predator is harvested provided the level of harvesting is controlled. Some examples are given to illustrate the behavior of the trajectories