IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Several approaches are proposed to solve global numerical optimization problems. Most of researchers have experimented the robustness of their algorithms by generating the result based on minimization aspect. In this paper, we focus on maximization problems by using several hybrid chemical reaction optimization algorithms including orthogonal chemical reaction optimization (OCRO), hybrid algorithm based on particle swarm and chemical reaction optimization (HP-CRO), real-coded chemical reaction optimization (RCCRO) and hybrid mutation chemical reaction optimization algorithm (MCRO), which showed success in minimization. The aim of this paper is to demonstrate that the approaches inspired by chemical reaction optimization are not only limited to minimization, but also are suitable for maximization. Moreover, experiment comparison related to other maximization algorithms is presented and discussed.
Complex dynamical behaviour of Disease Spread in a PlantHerbivore System with...iosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Mathematical modelling of Fish Resources Harvesting with Predator at Maximum Sustainable Yield
Kinfe Hailemariam Hntsaa, Zenebe Taka Mengesha (PhD)b*
aDepartment of Mathematics, Aksum University, Axum, Ethiopia, kinfhail@gmail.com
bDepartment of Biology, Aksum University, Axum, Ethiopia, zenebeteka2007@gmail.com
In this study, the population dynamic of fish is considered following Logistic model with the inclusion of harvesting. The prey-predator interaction is also considered with an assumption that the predator population which is completely theoretical and not physically defined has a little effect on the growth of prey population provided that there are no limiting factors other than the predators. This is to say that the prey-predator cycle remains stable as far as other factors are constant in the natural environment. The growth function of the predator population is constructed corresponding to the prey population, and its results showed that the predator population size is either convergent to a finite positive limit, zero or diverges to positive infinity; while the fish population size follows Logistic function and grows to an upper asymptote. Furthermore, the prey-predator interaction is considered with the assumption that the predator population has an effect on the growth of the prey population and the predator population has intra-specific competition for a limited environmental resource. Its result showed that the predator population size did not go to infinity without bound. In both cases the maximum sustainable yield is obtained, numerical simulation and stability analysis of the model are included.
Alternative Control of Insect Pests Using Paper in Corn Plantsinventionjournals
Vegetables are a very important source of food for humans due to their nutritious content such as fibre, vitamins, and minerals. In meeting people’s demand of vegetables, farmers have to deal with a number of obstacles, one of which is the problem caused by insect pests. A number of measures have been undertaken by farmers to control the insect pests. The aim of this research was to test the insect trapping devices made of coloured paper and lights. The colours tested were red, yellow, and green. This research was conducted from July to August 2016 in the experiment garden of SekolahMenengahKejuruan PembangunanPertanian (SMKPP), LembahSeulawahSub-district, Aceh Besar District, Aceh Province. The object of the research was corn plants.The method used wasRCBD with 3 treatments and 4 replicates. The observation variable was the insect population trapped in the coloured paper and lights. The finding of this research revealed that the yellow paper and light trapping devices captured the highest number of insects, followed by the paper and light traps of green and red colours.
Simulation modeling of the sensitivity analysis of differential effects of th...IJAEMSJORNAL
The vulnerability of the differential effects of the intrinsic growth rates of the fish population on the uncertainty analysis can only be controlled by using the mathematical technique of a sensitivity analysis that is called a local minimum selection method based on a Matlab numerical scheme of ordinary differential equations of order 45 (ODE 45). The quantification of the p-norms sensitivity analysis depends on the application of the 1-norm, 2-norm, 3-norm, 4-norm, 5-norm, 6-norm and infinity-norm. In the context of this study, the best-fit intrinsic growth rate of fish population with a small error has occurred when its value is 0.303 which minimizes the bigger sensitivity values previously obtained irrespective of the p-norm sensitivity values. The novel results which we have obtained have not been seen elsewhere. These results are fully presented and discussed in this study.
Speciation And Physicochemical Studies of Some Biospecific CompoundsIOSR Journals
Abstract: A green, safer , efficient , eco-friendly approach for the synthesis of novel compounds which reveal biological and spermicidal activity. The nature of the pharmacophore decides the physiological reactivity of the compound.
Several approaches are proposed to solve global numerical optimization problems. Most of researchers have experimented the robustness of their algorithms by generating the result based on minimization aspect. In this paper, we focus on maximization problems by using several hybrid chemical reaction optimization algorithms including orthogonal chemical reaction optimization (OCRO), hybrid algorithm based on particle swarm and chemical reaction optimization (HP-CRO), real-coded chemical reaction optimization (RCCRO) and hybrid mutation chemical reaction optimization algorithm (MCRO), which showed success in minimization. The aim of this paper is to demonstrate that the approaches inspired by chemical reaction optimization are not only limited to minimization, but also are suitable for maximization. Moreover, experiment comparison related to other maximization algorithms is presented and discussed.
Complex dynamical behaviour of Disease Spread in a PlantHerbivore System with...iosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Mathematical modelling of Fish Resources Harvesting with Predator at Maximum Sustainable Yield
Kinfe Hailemariam Hntsaa, Zenebe Taka Mengesha (PhD)b*
aDepartment of Mathematics, Aksum University, Axum, Ethiopia, kinfhail@gmail.com
bDepartment of Biology, Aksum University, Axum, Ethiopia, zenebeteka2007@gmail.com
In this study, the population dynamic of fish is considered following Logistic model with the inclusion of harvesting. The prey-predator interaction is also considered with an assumption that the predator population which is completely theoretical and not physically defined has a little effect on the growth of prey population provided that there are no limiting factors other than the predators. This is to say that the prey-predator cycle remains stable as far as other factors are constant in the natural environment. The growth function of the predator population is constructed corresponding to the prey population, and its results showed that the predator population size is either convergent to a finite positive limit, zero or diverges to positive infinity; while the fish population size follows Logistic function and grows to an upper asymptote. Furthermore, the prey-predator interaction is considered with the assumption that the predator population has an effect on the growth of the prey population and the predator population has intra-specific competition for a limited environmental resource. Its result showed that the predator population size did not go to infinity without bound. In both cases the maximum sustainable yield is obtained, numerical simulation and stability analysis of the model are included.
Alternative Control of Insect Pests Using Paper in Corn Plantsinventionjournals
Vegetables are a very important source of food for humans due to their nutritious content such as fibre, vitamins, and minerals. In meeting people’s demand of vegetables, farmers have to deal with a number of obstacles, one of which is the problem caused by insect pests. A number of measures have been undertaken by farmers to control the insect pests. The aim of this research was to test the insect trapping devices made of coloured paper and lights. The colours tested were red, yellow, and green. This research was conducted from July to August 2016 in the experiment garden of SekolahMenengahKejuruan PembangunanPertanian (SMKPP), LembahSeulawahSub-district, Aceh Besar District, Aceh Province. The object of the research was corn plants.The method used wasRCBD with 3 treatments and 4 replicates. The observation variable was the insect population trapped in the coloured paper and lights. The finding of this research revealed that the yellow paper and light trapping devices captured the highest number of insects, followed by the paper and light traps of green and red colours.
Simulation modeling of the sensitivity analysis of differential effects of th...IJAEMSJORNAL
The vulnerability of the differential effects of the intrinsic growth rates of the fish population on the uncertainty analysis can only be controlled by using the mathematical technique of a sensitivity analysis that is called a local minimum selection method based on a Matlab numerical scheme of ordinary differential equations of order 45 (ODE 45). The quantification of the p-norms sensitivity analysis depends on the application of the 1-norm, 2-norm, 3-norm, 4-norm, 5-norm, 6-norm and infinity-norm. In the context of this study, the best-fit intrinsic growth rate of fish population with a small error has occurred when its value is 0.303 which minimizes the bigger sensitivity values previously obtained irrespective of the p-norm sensitivity values. The novel results which we have obtained have not been seen elsewhere. These results are fully presented and discussed in this study.
Speciation And Physicochemical Studies of Some Biospecific CompoundsIOSR Journals
Abstract: A green, safer , efficient , eco-friendly approach for the synthesis of novel compounds which reveal biological and spermicidal activity. The nature of the pharmacophore decides the physiological reactivity of the compound.
Some Concepts on Constant Interval Valued Intuitionistic Fuzzy Graphsiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
A set D ? V of a given fuzzy graph G(V,?,µ) is a dominating set if for every u?V-D there exists v ?D such that (u,v) is a strong arc and ?(u)=?(v) and if the number of vertices of D is minimum then it is called a minimum dominating set of G. Domination number of G is the sum of membership values of vertices of a minimum dominating set D and it is denoted by ?(D). In this paper we study domination in fuzzy graphs. Also we formulate an algorithm to find dominating set for a given fuzzy graph.
Effect of mow procedure on physiological and biochemical properties of blood ...iosrjce
IOSR Journal of Biotechnology and Biochemistry (IOSR-JBB) covers studies of the chemical processes in living organisms, structure and function of cellular components such as proteins, carbohydrates, lipids, nucleic acids and other biomolecules, chemical properties of important biological molecules, like proteins, in particular the chemistry of enzyme-catalyzed reactions, genetic code (DNA, RNA), protein synthesis, cell membrane transport, and signal transduction. IOSR-JBB is privileged to focus on a wide range of biotechnology as well as high quality articles on genetic engineering, cell and tissue culture technologies, genetics, microbiology, molecular biology, biochemistry, embryology, cell biology, chemical engineering, bioprocess engineering, information technology, biorobotics.
The Dynamics of Sokol-Howell Prey-Predator Model Involving Strong Allee Effec...Saad Al-Momen
In this paper, a Sokol-Howell prey-predator model involving strong Allee effect is proposed and analyzed. The existence, uniqueness, and boundedness are studied. All the five possible equilibria have been are obtained and their local stability conditions are established. Using Sotomayor's theorem, the conditions of local saddle-node and transcritical and pitchfork bifurcation are derived and drawn. Numerical simulations are performed to clarify the analytical results.
Dynamical System of Tuberculosis Considering Lost Sight CompartmentDr. Amarjeet Singh
In this study, a model for the tuberculosis infection
considering vaccination and lost-sight compartement is
formulated. there are six populations in this model,
Susceptibled, vaccinated, exposed, lost sight, infected, and
recovered. The lost sight populations are infected but do not
get any treatment and still can spread the tuberculosis, the
infected population are infected but already got a treatment
and no longer spread the tuberculosis. The local stability are
obtained by analyzing the epidemic threshold ℛ0
. The result
shows that the disease-free equilibrium is locally
asymptotically stable when the condition ℛ0<1 is satisfied, and
the unique endemic equilibrium exist and it is locally
asymptotically stable if ℛ0>1 is satisfied. The numerical
simulation are also performed to support the analytical result.
Some Concepts on Constant Interval Valued Intuitionistic Fuzzy Graphsiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
A set D ? V of a given fuzzy graph G(V,?,µ) is a dominating set if for every u?V-D there exists v ?D such that (u,v) is a strong arc and ?(u)=?(v) and if the number of vertices of D is minimum then it is called a minimum dominating set of G. Domination number of G is the sum of membership values of vertices of a minimum dominating set D and it is denoted by ?(D). In this paper we study domination in fuzzy graphs. Also we formulate an algorithm to find dominating set for a given fuzzy graph.
Effect of mow procedure on physiological and biochemical properties of blood ...iosrjce
IOSR Journal of Biotechnology and Biochemistry (IOSR-JBB) covers studies of the chemical processes in living organisms, structure and function of cellular components such as proteins, carbohydrates, lipids, nucleic acids and other biomolecules, chemical properties of important biological molecules, like proteins, in particular the chemistry of enzyme-catalyzed reactions, genetic code (DNA, RNA), protein synthesis, cell membrane transport, and signal transduction. IOSR-JBB is privileged to focus on a wide range of biotechnology as well as high quality articles on genetic engineering, cell and tissue culture technologies, genetics, microbiology, molecular biology, biochemistry, embryology, cell biology, chemical engineering, bioprocess engineering, information technology, biorobotics.
The Dynamics of Sokol-Howell Prey-Predator Model Involving Strong Allee Effec...Saad Al-Momen
In this paper, a Sokol-Howell prey-predator model involving strong Allee effect is proposed and analyzed. The existence, uniqueness, and boundedness are studied. All the five possible equilibria have been are obtained and their local stability conditions are established. Using Sotomayor's theorem, the conditions of local saddle-node and transcritical and pitchfork bifurcation are derived and drawn. Numerical simulations are performed to clarify the analytical results.
Dynamical System of Tuberculosis Considering Lost Sight CompartmentDr. Amarjeet Singh
In this study, a model for the tuberculosis infection
considering vaccination and lost-sight compartement is
formulated. there are six populations in this model,
Susceptibled, vaccinated, exposed, lost sight, infected, and
recovered. The lost sight populations are infected but do not
get any treatment and still can spread the tuberculosis, the
infected population are infected but already got a treatment
and no longer spread the tuberculosis. The local stability are
obtained by analyzing the epidemic threshold ℛ0
. The result
shows that the disease-free equilibrium is locally
asymptotically stable when the condition ℛ0<1 is satisfied, and
the unique endemic equilibrium exist and it is locally
asymptotically stable if ℛ0>1 is satisfied. The numerical
simulation are also performed to support the analytical result.
Our journal has been transition marks a significant shift in the landscape of academic publishing, aiming to provide numerous benefits to both authors and readers. Open access has the potential to democratize knowledge, enhance research impact, and foster greater collaboration within the academic community.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
On Stability Equilibrium Analysis of Endemic MalariaIOSR Journals
This paper considers the stability equilibrium of malaria in a varying population. We established the
Disease free equilibrium and the endemic equilibrium and carried out the stability analysis of the Disease free
equilibrium state (the state of complete eradication of malaria from the population).
Deterministic Stabilization of a Dynamical System using a Computational ApproachIJAEMSJORNAL
The qualitative behavior of a multi-parameter dynamical system has been investigated. It is shown that changes in the initial data of a dynamical system will affect the stabilization of the steady-state solution which is originally unstable. It is further shown that the stabilization of a five-dimensional dynamical system can be used as an alternative method of verifying qualitatively the concept of the stability of a unique positive steady-state solution. These novel contributions have not been seen elsewhere; these are presented and discussed in this paper.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology
ESSENTIAL MODIFICATIONS ON BIOGEOGRAPHY-BASED OPTIMIZATION ALGORITHMcscpconf
Biogeography-based optimization (BBO) is a new population-based evolutionary algorithm and is based on an old theory of island biogeography that explains the geographical distribution of biological organisms. BBO was introduced in 2008 and then a lot of modifications were employed to enhance its performance. This paper proposes two modifications; firstly, modifying the probabilistic selection process of the migration and mutation stages to give a fairly randomized selection for all the features of the islands. Secondly, the clear duplication process after the mutation stage is sized to avoid any corruption on the suitability index variables. The obtained results through wide variety range of test functions with different dimensions and complexities proved that the BBO performance can be enhanced effectively
without using any complicated form of the immigration and emigration rates. This essential modification has to be considered as an initial step for any other modification.
Heterogeneity in biological populations, from cancer to ecological systems, is ubiquitous. Despite this knowledge, current mathematical models in population biology often do
not account for inter-individual heterogeneity. In systems such as cancer, this means assuming cellular homogeneity and deterministic phenotypes, despite the fact that heterogeneity is thought to play a role in therapy resistance. Glioblastoma Multiforme (GBM) is an aggressive and fatal form of brain cancer notoriously difficult to predict and treat due to its heterogeneous nature. In this talk, I will discuss several approaches I have developed towards incorporating and
estimating cellular heterogeneity in partial differential equation (PDE) models of GBM growth.
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC BAO AND HYPERCHAOTIC XU SYSTEMS VIA ACTI...IJCSEIT Journal
This paper investigates the anti-synchronization of identical hyperchaotic Bao systems (Bao and Liu,
2008), identical hyperchaotic Xu systems (Xu, Cai and Zheng, 2009) and non-identical hyperchaotic Bao
and hyperchaotic Xu systems. Active nonlinear control has been deployed for the anti- synchronization of
the hyperchaotic systems addressed in this paper and the main results have been established using
Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active
nonlinear control method is very effective and convenient to achieve anti-synchronization of identical and
non-identical hyperchaotic Bao and hyperchaotic Xu systems. Numerical simulations have been provided to
validate and demonstrate the effectiveness of the anti-synchronization results for the hyperchaotic Cao and
hyperchaotic Xu systems.
A System of Estimators of the Population Mean under Two-Phase Sampling in Pre...Premier Publishers
This paper deals with estimation of the population mean under two-phase sampling. Utilizing information on two-auxiliary variables, a system of estimators for estimating the finite population mean is proposed and its properties, up to the first order of approximation, are studied. As particular cases various estimators are suggested. The performance of suggested estimators is compared with some contemporary estimators of the population mean through numerical illustrations carried over the data set of some natural populations. Also, a small-scale Monte Carlo simulation is carried out for the empirical comparison.
HYBRID SYNCHRONIZATION OF LIU AND LÜ CHAOTIC SYSTEMS VIA ADAPTIVE CONTROLijait
This paper derives new results for the hybrid synchronization of identical Liu systems, identical Lü systems, and non-identical Liu and Lü systems via adaptive control method. Liu system (Liu et al. 2004) and Lü system (Lü and Chen, 2002) are important models of three-dimensional chaotic systems. Hybrid synchronization of the three-dimensional chaotic systems addressed in this paper is achieved through the synchronization of the first and last pairs of states and anti-synchronization of the middle pairs of the two systems. Adaptive control method is deployed in this paper for the general case when the system
parameters are unknown. Sufficient conditions for hybrid synchronization of identical Liu systems, identical Lü systems and non-identical Liu and Lü systems are derived via adaptive control theory and Lyapunov stability theory. Since the Lyapunov exponents are not needed for these calculations, the
adaptive control method is very effective and convenient for the hybrid synchronization of the chaotic systems addressed in this paper. Numerical simulations are shown to illustrate the effectiveness of the proposed synchronization schemes.
A Moment Inequality for Overall Decreasing Life Class of Life Distributions w...inventionjournals
:A moment inequality is derived for the system whose life distribution is in an overall decreasing life (ODL) class of life distributions. A new nonparametric test statistic for testing exponentiality against ODL is investigated based on this inequality. The asymptotic normality of the proposed statistic is presented. Pitman's asymptotic efficiency, power and critical values of this test are calculated to assess the performance of the test. Real examples are given to elucidate the use of the proposed test statistic in the reliability analysis. Wealso proposed a test for testing exponentiality versus ODL for right censored data and the power estimates of this test are also simulated for censored data for some commonly used distributions in reliability. Finally, real data are used as an example for practical problems.
The Ecology and Ecosystems unit notes blend in Evolutionary adaptations and Evidence for Evolution along with standard E&E topics. The notes has examples of key content areas. Originally designed for Junior High and High School students, we use these for 7-8th grade students and warm ups for High School students. Includes basic concepts, food chain, webs, energy pyramids, matter cyclers, predator-prey, trophic levels; along with Ecosystem types, carrying capacity as well as Rules of the Environment.
Similar to Modeling the Simultaneous Effect of Two Toxicants Causing Deformity in a Subclass of Biological Population (20)
An Examination of Effectuation Dimension as Financing Practice of Small and M...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Does Goods and Services Tax (GST) Leads to Indian Economic Development?iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Childhood Factors that influence success in later lifeiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Emotional Intelligence and Work Performance Relationship: A Study on Sales Pe...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Customer’s Acceptance of Internet Banking in Dubaiiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
A Study of Employee Satisfaction relating to Job Security & Working Hours amo...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Consumer Perspectives on Brand Preference: A Choice Based Model Approachiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Student`S Approach towards Social Network Sitesiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Broadcast Management in Nigeria: The systems approach as an imperativeiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
A Study on Retailer’s Perception on Soya Products with Special Reference to T...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
A Study Factors Influence on Organisation Citizenship Behaviour in Corporate ...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Consumers’ Behaviour on Sony Xperia: A Case Study on Bangladeshiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Design of a Balanced Scorecard on Nonprofit Organizations (Study on Yayasan P...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Public Sector Reforms and Outsourcing Services in Nigeria: An Empirical Evalu...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Media Innovations and its Impact on Brand awareness & Considerationiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Customer experience in supermarkets and hypermarkets – A comparative studyiosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Social Media and Small Businesses: A Combinational Strategic Approach under t...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Secretarial Performance and the Gender Question (A Study of Selected Tertiary...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Implementation of Quality Management principles at Zimbabwe Open University (...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Organizational Conflicts Management In Selected Organizaions In Lagos State, ...iosrjce
IOSR Journal of Business and Management (IOSR-JBM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of business and managemant and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications inbusiness and management. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Phenomics assisted breeding in crop improvementIshaGoswami9
As the population is increasing and will reach about 9 billion upto 2050. Also due to climate change, it is difficult to meet the food requirement of such a large population. Facing the challenges presented by resource shortages, climate
change, and increasing global population, crop yield and quality need to be improved in a sustainable way over the coming decades. Genetic improvement by breeding is the best way to increase crop productivity. With the rapid progression of functional
genomics, an increasing number of crop genomes have been sequenced and dozens of genes influencing key agronomic traits have been identified. However, current genome sequence information has not been adequately exploited for understanding
the complex characteristics of multiple gene, owing to a lack of crop phenotypic data. Efficient, automatic, and accurate technologies and platforms that can capture phenotypic data that can
be linked to genomics information for crop improvement at all growth stages have become as important as genotyping. Thus,
high-throughput phenotyping has become the major bottleneck restricting crop breeding. Plant phenomics has been defined as the high-throughput, accurate acquisition and analysis of multi-dimensional phenotypes
during crop growing stages at the organism level, including the cell, tissue, organ, individual plant, plot, and field levels. With the rapid development of novel sensors, imaging technology,
and analysis methods, numerous infrastructure platforms have been developed for phenotyping.
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
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mô tả các thí nghiệm về đánh giá tác động dòng khí hóa sau đốt
Modeling the Simultaneous Effect of Two Toxicants Causing Deformity in a Subclass of Biological Population
1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 6 Ver. IV (Nov. - Dec. 2015), PP 70-82
www.iosrjournals.org
DOI: 10.9790/5728-11647082 www.iosrjournals.org 70 | Page
Modeling the Simultaneous Effect of Two Toxicants Causing
Deformity in a Subclass of Biological Population
Anuj Kumar1
, A.W. Khan1
, A.K.Agrawal2
1
(Department of Mathematics, Integral University, Lucknow, India)
2
(Department of Mathematics, Amity Univesity, Lucknow, India)
Abstract: In this paper, we have proposed and analyzed a mathematical model to study the simultaneous effect
of two toxicants on a biological population, in which a subclass of biological population is severely affected and
exhibits abnormal symptoms like deformity, fecundity, necrosis, etc. On studying the qualitative behavior of
model, it is shown that the density of total population will settle down to an equilibrium level lower than the
carrying capacity of the environment. In the model, we have assumed that a subclass of biological population is
not capable in further reproduction and it is found that the density of this subclass increases as emission rates
of toxicants or uptake rates of toxicants increase. For large emission rates it may happen that the entire
population gets severely affected and is not capable in reproduction and after a time period all the population
may die out. The stability analysis of the model is determined by variational matrix and method of Lyapunov’s
function. Numerical simulation is given to illustrate the qualitative behavior of model.
Keywords: Biological species, Deformity, Mathematical model, Stability, Two toxicants.
AMS Classification – 93A30, 92D25, 34D20, 34C60
I. Introduction
The dynamics of effect of toxicants on biological species using mathematical models ([1], [2], [3], [4],
[5], [6], [7], [8], [9], [10], [11]) have been studied by many researchers. These studies have been carried out for
different cases such as: Rescigno ([9]) proposed a mathematical model to study the effect of a toxicant on a
biological species when toxicant is being produced by the species itself, Hallam et. al.([6], [7]) proposed and
analyzed a mathematical model to study the effect of a toxicant on the growth rate of biological species, Shukla
et. al. [11] proposed a model to study the simultaneous effect of two different toxicants, emitted from some
external sources, etc. In all of these studies, it is assumed that the toxicants affect each and every individual of
the biological species uniformly. But it is observed that some members of biological species get severely
affected by toxicants and show change in shape, size, deformity, etc. These changes are observed in the
biological species living in aquatic environment ([12], [13], [14], [15], [16], [17], [18], [19], [20]) and in
terrestrial environment, in plants ([21], [22]) and in animals ([23], [24], [25], [26]).
The study of such very important observable fact where a subclass of the biological species is
adversely affected by the toxicant and shows abnormal symptoms such as deformity, incapable in reproduction
etc. using mathematical models is very limited. Agrawal and Shukla [2] have studied the effect of a single
toxicant (emitted from some external sources) on a biological population in which a subclass of biological
population is severely affected and shows abnormal symptoms like deformity, fecundity, necrosis, etc. using
mathematical model. However, no study has been done for this phenomenon under the simultaneous effect of
two toxicants. Therefore, in this paper we have proposed a dynamical model to study the simultaneous effect of
two toxicants (both toxicants are constantly emitted from some external sources) on a biological species in
which a subclass of biological population is severely affected and shows abnormal symptoms like deformity,
fecundity, necrosis, etc.
II. Mathematical Model
We consider a logistically growing biological population with density 𝑁(𝑡) in the environment and
simultaneously affected by two different types of toxicants with environment concentrations 𝑇1 𝑡 and 𝑇2(𝑡)
(both toxicants are constantly emitted in the environment at the rates 𝑄1 and 𝑄2 respectively, from some
2. Modeling the Simultaneous Effect of Two Toxicants Causing Deformity in a Subclass of Biological…
DOI: 10.9790/5728-11647082 www.iosrjournals.org 71 | Page
external sources). These toxicants are correspondingly uptaken by the biological population at different
concentration rates 𝑈1 𝑡 and 𝑈2 𝑡 . These toxicants decrease the growth rate of biological population as well
as they also adversely affect a subclass of biological population with density 𝑁 𝐷(𝑡) and decay the capability of
reproduction. Here, 𝑁𝐴 𝑡 is the density of biological population which is capable in reproduction. Keeping
these views in mind, we have proposed the following model:
𝑑𝑁𝐴
𝑑𝑡
= 𝑏 − 𝑑 𝑁𝐴 − 𝑟1 𝑈1 + 𝑟2 𝑈2 𝑁𝐴 −
𝑟𝑁𝐴 𝑁
𝐾 𝑇1, 𝑇2
𝑑𝑁 𝐷
𝑑𝑡
= 𝑟1 𝑈1 + 𝑟2 𝑈2 𝑁𝐴 −
𝑟𝑁 𝐷 𝑁
𝐾 𝑇1, 𝑇2
− 𝛼 + 𝑑 𝑁 𝐷
𝑑𝑇1
𝑑𝑡
= 𝑄1 − 𝛿1 𝑇1 − 𝛾1 𝑇1 𝑁 + 𝜋1 𝜈1 𝑁𝑈1 2.1
𝑑𝑇2
𝑑𝑡
= 𝑄2 − 𝛿2 𝑇2 − 𝛾2 𝑇2 𝑁 + 𝜋2 𝜈2 𝑁𝑈2
𝑑𝑈1
𝑑𝑡
= 𝛾1 𝑇1 𝑁 − 𝛽1 𝑈1 − 𝜈1 𝑁𝑈1
𝑑𝑈2
𝑑𝑡
= 𝛾2 𝑇2 𝑁 − 𝛽2 𝑈2 − 𝜈2 𝑁𝑈2
𝑁𝐴 0 , 𝑁 𝐷 0 ≥ 0, 𝑇𝑖 0 ≥ 0, 𝑈𝑖 0 ≥ 𝑐𝑖 𝑁 0 , 𝑐𝑖 > 0, 0 < 𝜋𝑖 < 1 for 𝑖 = 1,2
All the parameters used in the model (2.1) are positive and defined as follows:
𝑏 − the birth rate of logistically growing biological population,
𝑑 − the death rate of logistically growing biological population,
𝑟 − the growth rate of biological population in toxicants free environment, i.e. 𝑟 = (𝑏 − 𝑑)
𝛼 − the decay rate of the deformed population due to high toxicity,
𝑟1 & 𝑟2 − the decreasing rates of the growth rate associated with the uptakes of environmental
concentration of toxicants 𝑇1 and 𝑇2 respectively,
𝛿1 & 𝛿2 − the natural depletion rate coefficients of 𝑇1 and 𝑇2 respectively,
𝛽1 & 𝛽2 − the natural depletion rate coefficients of 𝑈1 and 𝑈2 respectively,
𝛾1 & 𝛾2 − the depletion rate coefficients due to uptake by the population respectively,
(𝑖. 𝑒. 𝛾1 𝑇1 𝑁 & 𝛾2 𝑇2 𝑁)
𝜈1 & 𝜈2 − the depletion rate coefficients of 𝑈1 and 𝑈2 respectively due to decay of some members of 𝑁,
(𝑖. 𝑒. 𝜈1 𝑁𝑈1 & 𝜈2 𝑁𝑈2)
𝜋1 & 𝜋2 − the fractions of the depletion of 𝑈1 and 𝑈2 respectively due to decay of some members of
𝑁 which may reenter into the environment, 𝑖. 𝑒. 𝜋1 𝜈1 𝑁𝑈1 & 𝜋2 𝜈2 𝑁𝑈2
In the above model (2.1), total density of logistically growing biological population 𝑁 is equal to the
sum of density of biological population without deformity 𝑁𝐴 and with deformity 𝑁 𝐷, 𝑖. 𝑒. 𝑁 = 𝑁𝐴 + 𝑁 𝐷 .
So, the above system can be written in terms of 𝑁, 𝑁 𝐷, 𝑇1, 𝑇2, 𝑈1and 𝑈2 as follows:
𝑑𝑁
𝑑𝑡
= 𝑟𝑁 −
𝑟𝑁2
𝐾 𝑇1, 𝑇2
− 𝛼 + 𝑏 𝑁 𝐷
𝑑𝑁 𝐷
𝑑𝑡
= 𝑟1 𝑈1 + 𝑟2 𝑈2 (𝑁 − 𝑁 𝐷) −
𝑟𝑁 𝐷 𝑁
𝐾 𝑇1, 𝑇2
− 𝛼 + 𝑑 𝑁 𝐷
𝑑𝑇1
𝑑𝑡
= 𝑄1 − 𝛿1 𝑇1 − 𝛾1 𝑇1 𝑁 + 𝜋1 𝜈1 𝑁𝑈1 2.2
𝑑𝑇2
𝑑𝑡
= 𝑄2 − 𝛿2 𝑇2 − 𝛾2 𝑇2 𝑁 + 𝜋2 𝜈2 𝑁𝑈2
𝑑𝑈1
𝑑𝑡
= 𝛾1 𝑇1 𝑁 − 𝛽1 𝑈1 − 𝜈1 𝑁𝑈1
𝑑𝑈2
𝑑𝑡
= 𝛾2 𝑇2 𝑁 − 𝛽2 𝑈2 − 𝜈2 𝑁𝑈2
3. Modeling the Simultaneous Effect of Two Toxicants Causing Deformity in a Subclass of Biological…
DOI: 10.9790/5728-11647082 www.iosrjournals.org 72 | Page
𝑁 0 ≥ 0, 𝑁 𝐷 0 ≥ 0, 𝑇𝑖 0 ≥ 0, 𝑈𝑖 0 ≥ 𝑐𝑖 𝑁 0 , 0 ≤ 𝜋𝑖 ≤ 1, for 𝑖 = 1,2
where 𝑐1, 𝑐2 > 0 are constants relating to the initial uptake concentration 𝑈𝑖 0 with the initial density
of biological population 𝑁(0).
In the model (2.2), the function 𝐾 𝑇1, 𝑇2 > 0 (for all values of 𝑇1 & 𝑇2) denotes the carrying capacity
of the environment for the biological population 𝑁 and it decreases when 𝑇1 or 𝑇2 or both increase.
we have,
initial carrying capacity, 𝐾0 = 𝐾 0, 0 and
𝜕𝐾
𝜕 𝑇 𝑖
< 0 for 𝑇𝑖 > 0, 𝑖 = 1,2 2.3
III. Equilibrium points and stability analysis
The model (2.2) has two non – negative equilibrium points 𝐸1 = 0, 0,
𝑄1
𝛿1
,
𝑄2
𝛿2
, 0, 0 and 𝐸2 =
(𝑁∗
, 𝑁 𝐷
∗
, 𝑇1
∗
, 𝑇2
∗
, 𝑈1
∗
, 𝑈2
∗
). It is obvious that equilibria 𝐸1 exist, hence existence of 𝐸1 is not discussed.
Existence of 𝑬 𝟐: The value of 𝑁∗
, 𝑁 𝐷
∗
, 𝑇1
∗
, 𝑇2
∗
, 𝑈1
∗
and 𝑈2
∗
are the positive solutions of the following system of
equations:
𝑁 =
1
𝑟
𝑟 − 𝑟1 𝑈1 − 𝑟2 𝑈2 𝐾 𝑇1, 𝑇2 3.1
𝑁 𝐷 =
𝑟1 𝑈1 + 𝑟2 𝑈2 𝑁𝐾 𝑇1, 𝑇2
𝑟𝑁 + 𝑟1 𝑈1 + 𝑟2 𝑈2 + 𝛼 + 𝑑 𝐾 𝑇1, 𝑇2
3.2
𝑇1 =
𝑄1 𝛽1 + 𝜈1 𝑁
𝑓1 𝑁
= 𝑔1 𝑁 3.3
𝑇2 =
𝑄2 𝛽2 + 𝜈2 𝑁
𝑓2 𝑁
= 𝑔2 𝑁 3.4
𝑈1 =
𝑄1 𝛾1 𝑁
𝑓1 𝑁
= 1 𝑁 3.5
𝑈2 =
𝑄2 𝛾2 𝑁
𝑓2 𝑁
= 2 𝑁 3.6
where, 𝑓1 𝑁 = 𝛿1 𝛽1 + 𝛾1 𝛽1 + 𝛿1 𝜈1 𝑁 + 𝛾1 𝜈1 1 − 𝜋1 𝑁2
3.7
𝑓2 𝑁 = 𝛿2 𝛽2 + 𝛾2 𝛽2 + 𝛿2 𝜈2 𝑁 + 𝛾2 𝜈2 1 − 𝜋2 𝑁2
(3.8)
Using equations (3.1-3.8), we can assume a function
𝐹 𝑁 = 𝑟𝑁 − 𝑟 − 𝑟1 1 𝑁 − 𝑟2 2 𝑁 𝐾 𝑔1 𝑁 , 𝑔2 𝑁 (3.9)
From (3.9), we can say that
𝐹 0 < 0 and 𝐹 𝐾0 > 0
this implies there must exist a root between 0 and 𝐾0 for the equation 𝐹 𝑁 = 0, says 𝑁∗
.
Uniqueness of 𝑬 𝟐:
For 𝑁∗
to be unique root of 𝐹 𝑁 = 0, we must have
𝑑𝐹
𝑑𝑁
= 𝑟 + 𝐾 𝑔1 𝑁 , 𝑔2 𝑁 𝑟1
𝑑1
𝑑𝑁
+ 𝑟2
𝑑2
𝑑𝑁
− 𝑟 − 𝑟1 1 𝑁 − 𝑟2 2 𝑁
𝜕𝐾
𝜕𝑇1
𝑑𝑔1
𝑑𝑁
+
𝜕𝐾
𝜕𝑇2
𝑑𝑔2
𝑑𝑁
> 0
where
𝑑1
𝑑𝑁
=
𝑄1 𝛾1
𝑓1
2
𝑁
𝛿1 𝛽1 − 𝛾1 𝜈1 1 − 𝜋1 𝑁2
(3.10)
𝑑2
𝑑𝑁
=
𝑄2 𝛾2
𝑓2
2
𝑁
𝛿2 𝛽2 − 𝛾2 𝜈2 1 − 𝜋2 𝑁2
(3.11)
4. Modeling the Simultaneous Effect of Two Toxicants Causing Deformity in a Subclass of Biological…
DOI: 10.9790/5728-11647082 www.iosrjournals.org 73 | Page
𝑑𝑔1
𝑑𝑁
= −
𝑄1 𝛾1
𝑓1
2
𝑁
𝛽1
2
+ 2𝛽1 𝜈1 1 − 𝜋1 𝑁 + 𝜈1
2
1 − 𝜋1 𝑁2
< 0 (3.12)
𝑑𝑔2
𝑑𝑁
= −
𝑄2 𝛾2
𝑓2
2
𝑁
𝛽2
2
+ 2𝛽2 𝜈2 1 − 𝜋2 𝑁 + 𝜈2
2
1 − 𝜋2 𝑁2
< 0 (3.13)
Since,
𝜕𝐾
𝜕𝑇1
,
𝜕𝐾
𝜕𝑇2
< 0 (from eq. (2.3)) and
𝑑𝑔1
𝑑𝑁
,
𝑑𝑔2
𝑑𝑁
< 0 (from eq. (3.12-3.13)), this implies that:
𝑟 − 𝑟1 1 𝑁 − 𝑟2 2 𝑁
𝜕𝐾
𝜕𝑇1
𝑑𝑔1
𝑑𝑁
+
𝜕𝐾
𝜕𝑇2
𝑑𝑔2
𝑑𝑁
> 0
then
𝑑𝐹
𝑑𝑁
> 0, only when
𝑟 + 𝐾 𝑔1 𝑁 , 𝑔2 𝑁 𝑟1
𝑑1
𝑑𝑁
+ 𝑟2
𝑑2
𝑑𝑁
> 𝑟 − 𝑟1 1 𝑁 − 𝑟2 2 𝑁
𝜕𝐾
𝜕𝑇1
𝑑𝑔1
𝑑𝑁
+
𝜕𝐾
𝜕𝑇2
𝑑𝑔2
𝑑𝑁
(3.14)
Hence, if the conditions (3.14) is satisfied, the root 𝑁∗
of 𝐹 𝑁 = 0 is unique and lower than the carrying
capacity of the environment.
After that, we can compute the value of 𝑁 𝐷
∗
, 𝑇1
∗
, 𝑇2
∗
, 𝑈1
∗
and 𝑈2
∗
with the help of 𝑁∗
and equations (3.2-3.8).
3.1 Local stability analysis
To study the local stability behavior of the equilibrium points 𝐸1 = 0, 0,
𝑄1
𝛿1
,
𝑄2
𝛿2
, 0, 0 and 𝐸2 =
(𝑁∗
, 𝑁 𝐷
∗
, 𝑇1
∗
, 𝑇2
∗
, 𝑈1
∗
, 𝑈2
∗
), we compute the variational matrices 𝑀1 and 𝑀2 corresponding to the equilibrium points
𝐸1 and 𝐸2 such as:
𝑀1 =
𝑟 −(𝛼 + 𝑏) 0 0 0 0
0 −(𝛼 + 𝑑) 0 0 0 0
−
𝛾1 𝑄1
𝛿1
0 −𝛿1 0 0 0
−
𝛾2 𝑄2
𝛿2
0 0 −𝛿2 0 0
𝛾1 𝑄1
𝛿1
0 0 0 −𝛽1 0
𝛾2 𝑄2
𝛿2
0 0 0 0 −𝛽2
From 𝑀1, it is obvious that 𝐸1 is a saddle point unstable locally only in the 𝑁 − direction and with
stable manifold locally in the 𝑁 𝐷 − 𝑇1 − 𝑇2 − 𝑈1 − 𝑈2 space.
And
𝑀2 =
−𝑟
2𝑁∗
𝐾 𝑇1
∗
, 𝑇2
∗ − 1 − 𝛼 + 𝑏 𝑟𝑁∗2
𝐾1 𝑇1
∗
, 𝑇2
∗
𝑟𝑁∗2
𝐾2 𝑇1
∗
, 𝑇2
∗
0 0
𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
−
𝑟𝑁 𝐷
∗
𝐾 𝑇1
∗
, 𝑇2
∗ − 𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
𝑁∗
𝑁 𝐷
∗ 𝑟𝑁∗
𝑁 𝐷
∗
𝐾1 𝑇1
∗
, 𝑇2
∗
𝑟𝑁∗
𝑁 𝐷
∗
𝐾2 𝑇1
∗
, 𝑇2
∗
𝑟1 𝑁∗
− 𝑁 𝐷
∗
𝑟2 𝑁∗
− 𝑁 𝐷
∗
−𝛾1 𝑇1
∗
+ 𝜋1 𝜈1 𝑈1
∗
0 − 𝛿1 + 𝛾1 𝑁∗
0 𝜋1 𝜈1 𝑁∗
0
−𝛾2 𝑇2
∗
+ 𝜋2 𝜈2 𝑈2
∗
0 0 −(𝛿2 + 𝛾2 𝑁∗
) 0 𝜋2 𝜈2 𝑁∗
𝛾1 𝑇1
∗
− 𝜈1 𝑈1
∗
0 𝛾1 𝑁∗
0 −(𝛽1 + 𝜈1 𝑁∗
) 0
𝛾2 𝑇2
∗
− 𝜈2 𝑈2
∗
0 0 𝛾2 𝑁∗
0 −(𝛽2 + 𝜈2 𝑁∗
)
Here,
𝐾1 𝑇1
∗
, 𝑇2
∗
=
1
𝐾2 𝑇1
∗
, 𝑇2
∗ .
𝜕𝐾
𝜕𝑇1 𝑇1
∗,𝑇2
∗
< 0 and 𝐾2 𝑇1
∗
, 𝑇2
∗
=
1
𝐾2 𝑇1
∗
, 𝑇2
∗ .
𝜕𝐾
𝜕𝑇2 𝑇1
∗,𝑇2
∗
< 0
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According to the Gershgorin’s disc, all the eigenvalues of variational matrix 𝑀2 are negative or having
negative real parts if
𝐾 𝑇1
∗
, 𝑇2
∗
< 2𝑁∗
(3.15)
𝛼 + 𝑏 + 𝑟𝑁∗2
𝐾1 𝑇1
∗
, 𝑇2
∗
+ 𝑟𝑁∗2
𝐾2 𝑇1
∗
, 𝑇2
∗
< 𝑟
2𝑁∗
𝐾 𝑇1
∗
, 𝑇2
∗ − 1 (3.16)
𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
−
𝑟𝑁 𝐷
∗
𝐾 𝑇1
∗
, 𝑇2
∗ + 𝑟𝑁∗
𝑁 𝐷
∗
𝐾1 𝑇1
∗
, 𝑇2
∗
+ 𝑟𝑁∗
𝑁 𝐷
∗
𝐾2 𝑇1
∗
, 𝑇2
∗
+ 𝑟1 𝑁∗
− 𝑁 𝐷
∗
+ 𝑟2 𝑁∗
− 𝑁 𝐷
∗
< 𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
𝑁∗
𝑁 𝐷
∗ (3.17)
−𝛾1 𝑇1
∗
+ 𝜋1 𝜈1 𝑈1
∗
+ 𝜋1 𝜈1 𝑁∗
< 𝛿1 + 𝛾1 𝑁∗
(3.18)
−𝛾2 𝑇2
∗
+ 𝜋2 𝜈2 𝑈2
∗
+ 𝜋2 𝜈2 𝑁∗
< 𝛿2 + 𝛾2 𝑁∗
(3.19)
𝛾1 𝑇1
∗
− 𝜈1 𝑈1
∗
+ 𝛾1 𝑁∗
< (𝛽1 + 𝜈1 𝑁∗
) (3.20)
𝛾2 𝑇2
∗
− 𝜈2
∗
𝑈2
∗
+ 𝛾2 𝑁∗
< 𝛽2 + 𝜈2 𝑁∗
(3.21)
Hence, we can state the following theorem.
Theorem 1: The equilibrium point 𝐸2 is locally asymptotically stable if the conditions (3.15-3.21) are satisfied.
3.2 Global stability analysis
To found a set of sufficient conditions for globally asymptotically stable behavior of the equilibria 𝐸2,
we need a lemma which establishes the region of attraction of 𝐸2.
Lemma 1: The region
Ω = 𝑁, 𝑁 𝐷, 𝑇1, 𝑇2, 𝑈1, 𝑈2 : 0 ≤ 𝑁 ≤ 𝐾0, 0 ≤ 𝑁 𝐷 ≤
𝑟1 + 𝑟2 𝑄1 + 𝑄2 𝐾0
𝑟1 + 𝑟2 𝑄1 + 𝑄2 + 𝛿 𝑚 𝛼 + 𝑑
,
0 ≤ 𝑇1 + 𝑇2 + 𝑈1 + 𝑈2 ≤
𝑄1 + 𝑄2
𝛿 𝑚
where 𝛿 𝑚 = min 𝛿1, 𝛿2, 𝛽1, 𝛽2
attracts all solution initiating in the interior of the positive orthant.
Proof: From the first equation of model (2.2),
we have,
𝑑𝑁
𝑑𝑡
≤ 𝑟𝑁 −
𝑟𝑁2
𝐾0
= 𝑟 1 −
𝑁
𝐾0
𝑁
Thus, limsup𝑡→∞ 𝑁 𝑡 ≤ 𝐾0.
From the last four equations of model (2.2),
we have,
𝑑𝑇1
𝑑𝑡
+
𝑑𝑇2
𝑑𝑡
+
𝑑𝑈1
𝑑𝑡
+
𝑑𝑈2
𝑑𝑡
= 𝑄1 + 𝑄2 − 𝛿1 𝑇1 + 𝛿2 𝑇2 + 𝛽1 𝑈1 + 𝛽2 𝑈2 − 1 − 𝜋1 𝜈1 𝑁𝑈1 − 1 − 𝜋2 𝜈2 𝑁𝑈2
≤ 𝑄1 + 𝑄2 − 𝛿 𝑚 𝑇1 + 𝑇2 + 𝑈1 + 𝑈2
where 𝛿 𝑚 = min(δ1, δ2, β1
, β2
)
Thus, lim sup
𝑡→∞
𝑇1 + 𝑇2 + 𝑈1 + 𝑈2 ≤
𝑄1 + 𝑄2
𝛿 𝑚
From the second equation of model (2.2),
we have,
𝑑𝑁 𝐷
𝑑𝑡
= 𝑟1 𝑈1 + 𝑟2 𝑈2 𝑁 − 𝑁 𝐷 −
𝑟𝑁 𝐷 𝑁
𝐾 𝑇1, 𝑇2
− 𝛼 + 𝑑 𝑁 𝐷
≤
𝑟1 + 𝑟2 𝑄1 + 𝑄2
𝛿 𝑚
𝐾0 − 𝑁 𝐷 − 𝛼 + 𝑑 𝑁 𝐷
Thus, lim sup𝑡→∞ 𝑁 𝐷 𝑡 ≤
𝑟1 + 𝑟2 𝑄1 + 𝑄2 𝐾0
𝑟1 + 𝑟2 𝑄1 + 𝑄2 + 𝛿 𝑚 𝛼 + 𝑑
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proving the lemma.□
The following theorem establishes global asymptotic stability conditions for the equilibrium point 𝐸2.
Theorem 2: Let 𝐾 𝑇 satisfies the following inequalities in Ω with the assumptions in equation (2.3):
𝐾 𝑚 ≤ 𝐾 𝑇 ≤ 𝐾0, 0 ≤ −
𝜕𝐾
𝜕𝑇1
𝑇1, 𝑇2 ≤ 𝜅1, 0 ≤ −
𝜕𝐾
𝜕𝑇2
𝑇1, 𝑇2 ≤ 𝜅2
where 𝐾 𝑚 , 𝜅1 & 𝜅2 are positive constants.
Then 𝐸2 is globally asymptotically stable with respect to all solutions initiating in the interior of the
positive orthant, if the following conditions hold in Ω:
𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
− 𝛼 + 𝑏 +
𝑟𝐾0 𝑟1 + 𝑟2 (𝑄1 + 𝑄2)
𝐾 𝑇1
∗
, 𝑇2
∗
𝑟1 + 𝑟2 𝑄1 + 𝑄2 + 𝛿 𝑚 𝛼 + 𝑑
2
<
4𝑟
25
(𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
)
𝑁∗
𝑁 𝐷
∗
2𝑁∗
𝐾 𝑇1
∗
, 𝑇2
∗ − 1 (3.22)
𝛾1 + 𝜋1 𝜈1
𝑄1 + 𝑄2
𝛿 𝑚
+
𝑟𝐾0
2
𝜅1
𝐾 𝑚
2
2
<
4𝑟
15
𝛿1 + 𝛾1 𝑁∗
2𝑁∗
𝐾 𝑇1
∗
, 𝑇2
∗ − 1 (3.23)
𝛾2 + 𝜋2 𝜈2
𝑄1 + 𝑄2
𝛿 𝑚
+
𝑟𝐾0
2
𝜅2
𝐾 𝑚
2
2
<
4𝑟
15
𝛿2 + 𝛾2 𝑁∗
2𝑁∗
𝐾 𝑇1
∗
, 𝑇2
∗ − 1 (3.24)
𝛾1 + 𝜈1
𝑄1 + 𝑄2
𝛿 𝑚
2
<
4𝑟
15
𝛽1 + 𝜈1 𝑁∗
2𝑁∗
𝐾 𝑇1
∗
, 𝑇2
∗ − 1 (3.25)
𝛾2 + 𝜈2
𝑄1 + 𝑄2
𝛿 𝑚
2
<
4𝑟
15
𝛽2 + 𝜈2 𝑁∗
2𝑁∗
𝐾 𝑇1
∗
, 𝑇2
∗ − 1 (3.26)
𝑟𝐾0 𝜅1 𝑟1 + 𝑟2 𝑄1 + 𝑄2
𝐾 𝑚
2 𝑟1 + 𝑟2 𝑄1 + 𝑄2 + 𝛼 + 𝑑 𝛿 𝑚
2
<
4
15
𝛿1 + 𝛾1 𝑁∗
𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
𝑁∗
𝑁 𝐷
∗ (3.27)
𝑟𝐾0 𝜅2 𝑟1 + 𝑟2 𝑄1 + 𝑄2
𝐾 𝑚
2 𝑟1 + 𝑟2 𝑄1 + 𝑄2 + 𝛼 + 𝑑 𝛿 𝑚
2
<
4
15
𝛿2 + 𝛾2 𝑁∗
𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
𝑁∗
𝑁 𝐷
∗ (3.28)
𝑟1 𝐾0
2
<
4
15
𝛽1 + 𝜈1 𝑁∗
𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
𝑁∗
𝑁 𝐷
∗ (3.29)
𝑟2 𝐾0
2
<
4
15
𝛽2 + 𝜈2 𝑁∗
𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
𝑁∗
𝑁 𝐷
∗ (3.30)
𝛾1 + 𝜋1 𝜈1 𝑁∗ 2
<
4
9
𝛿1 + 𝛾1 𝑁∗
𝛽1 + 𝜈1 𝑁∗
(3.31)
𝛾2 + 𝜋2 𝜈2 𝑁∗ 2
<
4
9
𝛿2 + 𝛾2 𝑁∗
𝛽2 + 𝜈2 𝑁∗
(3.32)
The proof of Theorem 2 is given in Appendix A.
IV. Numerical simulation
To make the qualitative results more clear, we give here numerical simulation of model (2.2) by
defining the function:
𝐾 𝑇1, 𝑇2 = 𝐾0 −
𝑏11 𝑇1
1 + 𝑏12 𝑇1
−
𝑏21 𝑇2
1 + 𝑏22 𝑇2
(4.1)
and assuming a set of parameters
𝑏 = 0.005, 𝑑 = 0.00001, 𝑟1 = 0.0007, 𝑟2 = 0.0005, 𝑄1 = 0.001, 𝑄2 = 0.0004
𝛿1 = 0.004, 𝛿2 = 0.001, 𝛾1 = 0.0005, 𝛾2 = 0.0003, 𝜋1 = 0.0004, 𝜋2 = 0.0006
𝜈1 = 0.005, 𝜈2 = 0.003, 𝛽1 = 0.006, 𝛽2 = 0.004, 𝐾0 = 10.0, 𝑏11 = 0.0002,
𝑏12 = 1.0, 𝑏21 = 0.0001, 𝑏22 = 2.0, 𝜅1 = 0.001, 𝜅2 = 0.001, 𝐾 𝑚 = 3.0
(4.2)
For the above function and set of values of parameters (4.1-4.2), we have obtained equilibrium point
𝐸2(𝑁∗
, 𝑁 𝐷
∗
, 𝑇1
∗
, 𝑇2
∗
, 𝑈1
∗
, 𝑈2
∗
) with values 𝑁∗
= 9.7771, 𝑁𝑑
∗
= 0.0227, 𝑇1
∗
= 0.1113, 𝑇2
∗
= 0.1002, 𝑈1
∗
= 0.0099
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DOI: 10.9790/5728-11647082 www.iosrjournals.org 76 | Page
and 𝑈2
∗
= 0.0088. Here, condition (3.14) satisfies which shows that the values 𝑁∗
, 𝑁 𝐷
∗
, 𝑇1
∗
, 𝑇2
∗
, 𝑈1
∗
and 𝑈2
∗
are
unique in the region Ω. The eigenvalues of variational matrix 𝑀2 corresponding to the equilibrium point 𝐸2 for
the model (2.2) are obtained as −0.0559, −0.0339, −0.0090, −0.0039, −0.0050 + 0.0002𝑖 and
−0.0050 − 0.0002𝑖. We note that four eigenvalues of variational matrix are negative and remaining two
eigenvalues have negative real parts which show that equilibrium point 𝐸2 is locally asymptotically stable. Also,
the equilibrium point 𝐸2 satisfies all the conditions of global asymptotic stability (3.22-3.32). (see Fig.1)
Fig.1: Nonlinear stability of (𝑵∗, 𝑵 𝑫
∗
) in 𝑵 − 𝑵 𝑫 plane for different initial starts
In Fig.2 & Fig.3, we have shown the changes in density of deformed population with respect to time
for different values of emission rates of toxicant in the environment 𝑄1 and 𝑄2 respectively. Here, we take all
the parameters same as eq. (4.2) except 𝑄1 and 𝑄2. In both figures, we can see that when emission rate of
toxicant 𝑄1 as well as emission rate of toxicant 𝑄2 increases the density of the deformed population also
increases, which shows that more members of the population will get deformed if the rate of toxicant emission
increases.
Fig.2: Variation of deformed population 𝑵 𝑫 with time for different values of 𝑸 𝟏
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DOI: 10.9790/5728-11647082 www.iosrjournals.org 77 | Page
Fig.3: Variation of deformed population 𝑵 𝑫 with time for different values of 𝑸 𝟐
In Fig.4 & Fig.5, we have represented the variation in the density of deformed population for different
values of the uptake rate coefficients 𝛾1 and 𝛾2 (all the parameters same as eq. (4.2) except 𝛾1 and 𝛾2
respectively). Here figures are showing that when the uptake rates of toxicants increase, density of deformed
population increases.
Fig.4: Variation of deformed population 𝑵 𝑫 with time for different values of 𝜸 𝟏
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DOI: 10.9790/5728-11647082 www.iosrjournals.org 78 | Page
Fig.5: Variation of deformed population 𝑵 𝑫 with time for different values of 𝜸 𝟐
In Fig.6, we have shown the variation in density of deformed population corresponding to the decay
rate of the deformed population due to high toxicity 𝛼 (all the parameters same as eq. (4.2) except 𝛼). In this
figure, we can see that when the decay rate of deformed population increases density of deformed population
decreases.
Fig.6: Variation of deformed population 𝑵 𝑫 with time for different values of 𝜶
Fig.7: 𝑵 and 𝑵 𝑫 for large emission rate of toxicant 𝑻 𝟏 in the environment
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DOI: 10.9790/5728-11647082 www.iosrjournals.org 79 | Page
Fig.8: 𝑵 and 𝑵 𝑫 for large emission rate of toxicant 𝑻 𝟐 in the environment
In Fig.7 & Fig.8, we have represented the variation in the densities of Total population (𝑁) and
Deformed population 𝑁 𝐷 for large emission rate of toxicants 𝑄1 and 𝑄2. These figures show that density of
total population gets severely affected and is not capable in reproduction for large emission rates.
V. Conclusion
In this paper, we have proposed and analyzed a mathematical model to study the simultaneous effect of
two toxicants on a biological population, in which a subclass of biological population is severely affected and
exhibits abnormal symptoms like deformity, fecundity, necrosis, etc. Here, we assume that these two toxicants
are being emitted into the environment by some external sources such as industrial discharge, vehicular exhaust,
waste water discharge from cities, etc. The model (2.2) has two equilibrium points 𝐸1 and 𝐸2 in which 𝐸1 is
saddle point and 𝐸2 is locally and globally stable under some conditions. The qualitative behavior of model (2.2)
shows that the density of total population will settle down to an equilibrium level, lower than its initial carrying
capacity. It is assumed that a subclass of biological population is not capable in reproduction. Under this
assumption, it is found that the density of this subclass increases as emission rates of toxicants or uptake rates of
toxicants increase and when the decay rate of deformed population increases, density of deformed population
decreases. For large emission rates, it may happen that the entire population gets severely affected and is not
capable in reproduction and after a time period all the population may die out. So, we need to control the
emission of toxicants from industries, household and vehicular discharges in the environment to protect
biological species from deformity.
Appendix A. Proof of the Theorem 2.
Proof: we consider a positive definite function about 𝐸2
𝑊 𝑁, 𝑁 𝐷, 𝑇1, 𝑇2, 𝑈1, 𝑈2
=
1
2
𝑁 − 𝑁∗ 2
+
1
2
𝑁 𝐷 − 𝑁 𝐷
∗ 2
+
1
2
𝑇1 − 𝑇1
∗ 2
+
1
2
𝑇2 − 𝑇2
∗ 2
+
1
2
𝑈1 − 𝑈1
∗ 2
+
1
2
𝑈2 − 𝑈2
∗ 2
Differentiating 𝑊 with respect to 𝑡 along the solution of (2.2), we get
𝑑𝑊
𝑑𝑡
= 𝑁 − 𝑁∗
𝑟𝑁 −
𝑟𝑁2
𝐾 𝑇1, 𝑇2
− 𝛼 + 𝑏 𝑁 𝐷
+ 𝑁 𝐷 − 𝑁 𝐷
∗
𝑟1 𝑈1 + 𝑟2 𝑈2 𝑁 − 𝑁 𝐷 −
𝑟𝑁 𝐷 𝑁
𝐾 𝑇1, 𝑇2
− 𝛼 + 𝑑 𝑁 𝐷
+ 𝑇1 − 𝑇1
∗
𝑄1 − 𝛿1 𝑇1 − 𝛾1 𝑇1 𝑁 + 𝜋1 𝜈1 𝑁𝑈1 + 𝑇2 − 𝑇2
∗
𝑄2 − 𝛿2 𝑇2 − 𝛾2 𝑇2 𝑁 + 𝜋2 𝜈2 𝑁𝑈2
+ 𝑈1 − 𝑈1
∗
𝛾1 𝑇1 𝑁 − 𝛽1 𝑈1 − 𝜈1 𝑁𝑈1 + 𝑈2 − 𝑈2
∗
𝛾2 𝑇2 𝑁 − 𝛽2 𝑈2 − 𝜈2 𝑁𝑈2
12. Modeling the Simultaneous Effect of Two Toxicants Causing Deformity in a Subclass of Biological…
DOI: 10.9790/5728-11647082 www.iosrjournals.org 81 | Page
𝑏12 = − 𝛼 + 𝑏 + 𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
−
𝑟𝑁 𝐷
𝐾 𝑇1
∗
, 𝑇2
∗ , 𝑏13 = 𝜋1 𝜈1 𝑈1 − 𝛾1 𝑇1 − 𝑟𝑁2
𝜂1 𝑇1, 𝑇2
𝑏14 = 𝜋2 𝜈2 𝑈2 − 𝛾2 𝑇2 − 𝑟𝑁2
𝜂2 𝑇1
∗
, 𝑇2 , 𝑏15 = 𝛾1 𝑇1 − 𝜈1 𝑈1 , 𝑏16 = 𝛾2 𝑇2 − 𝜈2 𝑈2
𝑏23 = −𝑟𝑁𝑁 𝐷 𝜂1 𝑇1, 𝑇2 , 𝑏24 = −𝑟𝑁𝑁 𝐷 𝜂2 𝑇1
∗
, 𝑇2 , 𝑏25 = 𝑟1 𝑁 − 𝑁 𝐷 , 𝑏26 = 𝑟2 𝑁 − 𝑁 𝐷
𝑏35 = 𝜋1 𝜈1 𝑁∗
+ 𝛾1 𝑁∗
, 𝑏46 = 𝜋2 𝜈2 𝑁∗
+ 𝛾2 𝑁∗
Thus,
dW
dt
will be negative definite provided
𝑏12
2
< 𝑏11 𝑏22 (3.33)
𝑏13
2
< 𝑏11 𝑏33 (3.34)
𝑏14
2
< 𝑏11 𝑏44 (3.35)
𝑏15
2
< 𝑏11 𝑏55 (3.36)
𝑏16
2
< 𝑏11 𝑏66 3.37
𝑏23
2
< 𝑏22 𝑏33 (3.38)
𝑏24
2
< 𝑏22 𝑏44 (3.39)
𝑏25
2
< 𝑏22 𝑏55 (3.40)
𝑏26
2
< 𝑏22 𝑏66 (3.41)
𝑏35
2
< 𝑏33 𝑏55 (3.42)
𝑏46
2
< 𝑏44 𝑏66 (3.43)
We note that (3.33-3.43) ⇒ (3.22-3.32) respectively. So, W is a Lyapunov’s function with respect to
the equilibrium 𝐸2 and therefore 𝐸2 is globally asymptotically stable under the conditions (3.22-3.32). Hence the
theorem. □
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