Continuous Models Logistic and Malthusian Growth
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Ecological Principles and Function of Natural Ecosystems by Professor Michel RICARD
Intensive Programme on Education for sustainable development in Protected Areas Amfissa, Greece, July 2014 ------------------------------------------------------------------------ Ecological principles and function of natural ecosystems By Professor Michel RICARD Summary 1. Hierarchy of living world 2. What is Ecology 3. The Biosphere - Lithosphere - Hydrosphere - Atmosphere 4. What is an ecosystem - Ecozone - Biome - Ecosystem - Ecological community - Habitat/biotope - Ecotone - Niche 5. Biological classification 6. Ecosystem processes - Radiation: heat, temperature and light - Primary production - Secondary production - Food web and trophic levels - Trophic cascade and ecology flow 7. Population ecology and population dynamics 8. Disturbance and resilience - Human impacts on resilience 9. Nutrient cycle, decomposition and mineralization - Nutrient cycle - Decomposition 10. Ecological amplitude 11. Ecology, environmental influences, biological interactions 12. Biodiversity 13. Environmental degradation - Water resources degradation - Climate change - Nutrient pollution - Eutrophication - Other examples of environmental degradation M. Ricard: Summer courses, Amfissa July 2014 1 1. Hierarchy of living world The larger objective of ecology is to understand the nature of environmental influences on individual organisms, populations, communities and ultimately at the level of the biosphere. If ecologists can achieve an understanding of these relationships, they will be well placed to contribute to the development of systems by which humans -
A Model for Bio-Economics of Fisheries
International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol. 2 Issue 2, February- 2013 A Model for Bio-Economics of Fisheries G. Shanmugam 1, K. B. Naidu 2 1Associate Professor, Dept of Mathematics, Jeppiaar Engineering College, Chennai, 2Professor, Department of Mathematics, Sathyabama University, Chennai Abstract. In this paper a model for growth of fish, a model for fishing economics and delay model for fishing are considered. The maximum sustainable yield for fishing is obtained. In the delay model the three cases of equilibrium population being equal to (or) greater then (or) less then the ratio of carrying capacity and rate of growth are considered. 1 Introduction The World population is growing at enormous rate, creating increasing demand for food. Food comes from renewable resources. Agricultural products are renewable resources, since every season new crops are produced in farms. Fisheries are a renewable resource since fish are reproduced in lakes and seas. Forests are renewable resources since they reproduce periodically. As these resources are renewable, the quality of the resources will certainly degrade, leading to shortage. Over fishing will lead to decline in fisheries. Global warming again has an impact on the growth of agriculture, fisheries and forests. It is imperative that we should manage these resources economically to prevent a catastrophic bust in our global economy. Mathematical bio-economics is the mathematical study of the management of renewable bio resources. It takes into consideration not only economic factors like revenue, cost etc., but also the impact of this demand on the resources. IJERTIJERT One of the mathematical tools used in bio economics is differential equations. -
Critical Review of the Literature on Marine Mammal Population Modelling Edward O
Nova Southeastern University NSUWorks Marine & Environmental Sciences Faculty Reports Department of Marine and Environmental Sciences 9-1-2008 Critical Review of the Literature on Marine Mammal Population Modelling Edward O. Keith Nova Southeastern University Oceanographic Center Find out more information about Nova Southeastern University and the Halmos College of Natural Sciences and Oceanography. Follow this and additional works at: https://nsuworks.nova.edu/occ_facreports Part of the Marine Biology Commons, and the Oceanography and Atmospheric Sciences and Meteorology Commons NSUWorks Citation Edward O. Keith. 2008. Critical Review of the Literature on Marine Mammal Population Modelling .E&P Sound & Marine Life Programme : 1 -63. https://nsuworks.nova.edu/occ_facreports/93. This Report is brought to you for free and open access by the Department of Marine and Environmental Sciences at NSUWorks. It has been accepted for inclusion in Marine & Environmental Sciences Faculty Reports by an authorized administrator of NSUWorks. For more information, please contact [email protected]. Critical Review of the Literature on Marine Mammal Population Modeling Prepared by Nova Southeastern University www.soundandmarinelife.org Revised Final Report Critical Review of the Literature on Marine Mammal Population Modeling (JIP 22-07-19) Edward O. Keith Oceanographic Center Nova Southeastern University 8000 N. Ocean Drive Dania Beach, FL 33004 [email protected] 954-262-8322 (voice) 954-262-4098 1 September 2008 1 Table of Contents A. Executive summary 3 B. Introduction 3 1. Purpose 4 2. Objectives 4 3. Scope 5 C. Types of population models 1. H0 vs. Model Selection/GLMs/GAMs 6 2. Exponential/Logistical 8 3. Multinomial 10 4. -
Diffusive Logistic Population Growth with Immigration
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Applied Mathematics Letters 18 (2005) 261–265 www.elsevier.com/locate/aml Diffusive logistic population growth with immigration S. Harris∗ College of Engineering and Applied Sciences and Marine Sciences Research Center, SUNY, Stony Brook, NY 11794, United States Received 1 March 2003; accepted 1 March 2003 Abstract We study diffusive logistic growth with immigration for a habitat surrounded by a hostile environment. The focus of our interest is the effect of immigration on the critical habitat length required for survival of the population. As expected, we find that this is reduced when immigration occurs. We also briefly consider the much simpler case where both immigration and emigration take place in an isolated habitat. © 2005 Elsevier Ltd. All rights reserved. Keywords: Diffusive logistic growth; Immigration; Critical length; Emigration 1. Introduction Various approaches have been taken to model the distribution of populations. These differ in several important respects, especially as regards the treatment of the dispersal process. Metapopulation models [1]provide a global description of a complex habitat whose evolution is determined by the colonization (i.e. immigration into) and extinction of its component patches. In contrast, reaction–diffusion models [2–4]provide a local description of individual isolated patches. The fusion of these two approaches would be desirable [5], and our purpose here is to consider one of the specific issues that must be resolved before such an ambitious goal can be reached. Existing results for reaction–diffusion models describing individual patches are almost exclusively limited to describing habitats in which there is no immigration. -
Projections of Global Carrying Capacity - Graeme Hugo
THE ROLE OF FOOD, AGRICULTURE, FORESTRY AND FISHERIES IN HUMAN NUTRITION – Vol. III - Projections of Global Carrying Capacity - Graeme Hugo PROJECTIONS OF GLOBAL CARRYING CAPACITY Graeme Hugo Professor of Geography, University of Adelaide, Australia Keywords: Carrying capacity, optimum population, population pressure, renewable resources, resource exploitation, economic development, fossil fuels, hunter-gatherers, ecological sustainability, cereal grains, consumption Contents 1. Introduction 2. The Reality of Projected Population Growth 3. Responses to Population Pressure on Resources 4. Optimum Populations 5. Food Production Outlook 6. Projections of Global Carrying Capacity 7. Conclusion Glossary Bibliography Biographical Sketch Summary The term carrying capacity is applied largely to animal populations as the maximum number of individuals in a particular species that can be indefinitely supported by the resources in a particular area. In animal contexts the carrying capacity is determined by the amount of food available, the number of predators, and the rate at which the environment can replace the resources that are used by the population. In applying this concept to humans there are two differences. First, human beings have the capacity to innovate and to use technology and to pass innovations on to future generations, so they have the capacity to redefine upward the limits imposed by carrying capacity. Second, human beings need and use a wider range of resources than food and water in the environment. Hence, human carrying capacity is a function of the resources in an area, the consumption level of those resources, and the technology used in exploitingUNESCO them. Therefore, there is –a grEOLSSeat deal of difficulty experienced in operationalizing or measuring human carrying capacity globally or regionally. -
Math 636 - Mathematical Modeling Discrete Modeling 1 Discrete Modeling – Population of the United States U
Discrete Modeling { Population of the United States Discrete Modeling { Population of the United States Variation in Growth Rate Variation in Growth Rate Autonomous Models Autonomous Models Outline Math 636 - Mathematical Modeling Discrete Modeling 1 Discrete Modeling { Population of the United States U. S. Population Malthusian Growth Model Programming Malthusian Growth Joseph M. Mahaffy, 2 Variation in Growth Rate [email protected] General Discrete Dynamical Population Model Linear Growth Rate U. S. Population Model Nonautonomous Malthusian Growth Model Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research Center 3 Autonomous Models San Diego State University Logistic Growth Model San Diego, CA 92182-7720 Beverton-Holt Model http://jmahaffy.sdsu.edu Analysis of Autonomous Models Fall 2018 Discrete Modeling U. S. Population | Discrete Modeling U. S. Population | Joseph M. Mahaffy, [email protected] (1/34) Joseph M. Mahaffy, [email protected] (2/34) Discrete Modeling { Population of the United States Discrete Modeling { Population of the United States Malthusian Growth Model Malthusian Growth Model Variation in Growth Rate Variation in Growth Rate Programming Malthusian Growth Programming Malthusian Growth Autonomous Models Autonomous Models United States Census Census Data Census Data United States Census 1790 3,929,214 1870 39,818,449 1950 150,697,361 Constitution requires census every 10 years 1800 5,308,483 1880 50,189,209 1960 179,323,175 Census used for budgeting federal payments and representation -
Carrying Capacity
CarryingCapacity_Sayre.indd Page 54 12/22/11 7:31 PM user-f494 /203/BER00002/Enc82404_disk1of1/933782404/Enc82404_pagefiles Carrying Capacity Carrying capacity has been used to assess the limits of into a single defi nition probably would be “the maximum a wide variety of things, environments, and systems to or optimal amount of a substance or organism (X ) that convey or sustain other things, organisms, or popula- can or should be conveyed or supported by some encom- tions. Four major types of carrying capacity can be dis- passing thing or environment (Y ).” But the extraordinary tinguished; all but one have proved empirically and breadth of the concept so defi ned renders it extremely theoretically fl awed because the embedded assump- vague. As the repetitive use of the word or suggests, car- tions of carrying capacity limit its usefulness to rying capacity can be applied to almost any relationship, bounded, relatively small-scale systems with high at almost any scale; it can be a maximum or an optimum, degrees of human control. a normative or a positive concept, inductively or deduc- tively derived. Better, then, to examine its historical ori- gins and various uses, which can be organized into four he concept of carrying capacity predates and in many principal types: (1) shipping and engineering, beginning T ways prefi gures the concept of sustainability. It has in the 1840s; (2) livestock and game management, begin- been used in a wide variety of disciplines and applica- ning in the 1870s; (3) population biology, beginning in tions, although it is now most strongly associated with the 1950s; and (4) debates about human population and issues of global human population. -
Smith's Population Model in a Slowly Varying Environment
Smith’s Population Model in a Slowly Varying Environment Rohit Kumar Supervisor: Associate Professor John Shepherd RMIT University CONTENTS 1. Introduction 2. Smith’s Model 3. Why was Smith’s Model proposed? 4. The Constant Coefficient Case 5. The Multi-Scaling Approach 6. The Multi-Scale Equation and its implicit solution using the Perturbation Approach 7. Smith’s parameter, c , varying slowly with time using Multi- Timing Approximations 8. Comparison of Multi-Timing Approximations with Numerical Solutions 9. Conclusion 1 1. Introduction Examples of the single species population models include the Malthusian model, Verhulst model and the Smith’s model, which will be the main one used in our project (Banks 1994). The Malthusian growth model named after Thomas Malthus illustrates the human population growing exponentially and deals with one positive parameter R , which is the intrinsic growth rate and one variable N (Banks 1994). The Malthusian growth model is defined by the initial-value problem dN RN, N(0) N . (1) dT 0 The Malthusian model (1) generated solutions NT()that were unbounded, which was very unrealistic and did not take into account populations that are limited in growth and hence, the Verhulst model was proposed (Bacaer 2011; Banks 1994). This model suggested that while the human population nurtured and doubled after some time, there comes a stage where it tends to steady state (Bacaer 2011). The Verhulst model is defined by the initial-value problem, dN N RN1 , N(0) N0 . (2) dT K Here, R is the intrinsic growth rate, N is the population size and K is the carrying capacity. -
Stochastic Predation Exposes Prey to Predator Pits and Local Extinction
130 300–309 OIKOS Research Stochastic predation exposes prey to predator pits and local extinction T. J. Clark, Jon S. Horne, Mark Hebblewhite and Angela D. Luis T. J. Clark (https://orcid.org/0000-0003-0115-3482) ✉ ([email protected]), M. Hebblewhite (https://orcid.org/0000-0001-5382-1361) and A. D. Luis, Wildlife Biology Program, Dept of Ecosystem and Conservation Sciences, W. A. Franke College of Forestry and Conservation, Univ. of Montana, Missoula, MT, USA. – J. S. Horne, Idaho Dept of Fish and Game, Lewiston, ID, USA. Oikos Understanding how predators affect prey populations is a fundamental goal for ecolo- 130: 300–309, 2021 gists and wildlife managers. A well-known example of regulation by predators is the doi: 10.1111/oik.07381 predator pit, where two alternative stable states exist and prey can be held at a low density equilibrium by predation if they are unable to pass the threshold needed to Subject Editor: James D. Roth attain a high density equilibrium. While empirical evidence for predator pits exists, Editor-in-Chief: Dries Bonte deterministic models of predator–prey dynamics with realistic parameters suggest they Accepted 6 November 2020 should not occur in these systems. Because stochasticity can fundamentally change the dynamics of deterministic models, we investigated if incorporating stochastic- ity in predation rates would change the dynamics of deterministic models and allow predator pits to emerge. Based on realistic parameters from an elk–wolf system, we found predator pits were predicted only when stochasticity was included in the model. Predator pits emerged in systems with highly stochastic predation and high carrying capacities, but as carrying capacity decreased, low density equilibria with a high likeli- hood of extinction became more prevalent. -
Mathematical Modeling for Population Projection and Management: a Case Study of Niger State
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 5 Ver. III (Sep. - Oct. 2017), PP 51-57 www.iosrjournals.org Mathematical Modeling for Population Projection and Management: A Case Study Of Niger State. Yahaya, Ahmad Abubakar1, Philip Moses Audu2 And Hassan Sheikh Aisha3 Department Of Mathematic, Federal Polytechnic, Bida. Niger State, Nigeria. Abstract: The purpose of this research paper is to develop a mathematical model to predict population figures of Niger state for the period of 20 years using2006 census figures. The data used were collected from National Population commission and were analyzed by using MATLAB Software. The idea behind the projection is to get an estimated figure of the population of Niger state without waiting for census data. The exponential growth model, also known as Malthus Model used in this paper simply shows that for any number of year specified as the proposed year for the growth model, the corresponding value can be obtain which importantly predicts the estimated amount of population. The parameter used for the population, can be used for any given region, which helps to check unwanted population increase or decrease. It can also be exploited to access the success of the method implemented over time. ----------------------------------------------------------------------------------------------------------------------------- ---------- Date of Submission: 21-10-2017 Date of acceptance: 27-10-2017 -------------------------------------------------------------------------------------------------------------------- ------------------- I. Introduction Projection of any country‟s population plays a significant role in the planning as well as in the decision making for the socio-economic and demographic development. The population projection has become one of the most important problems in the world. -
Introduction
1 Introduction Many of the principles underlying the behavior of the natural world are statements and relations involving rates at which things happen. When they are expressed in mathematical terms the relations are equations and the rates are derivatives. Equations containing derivatives are differential equations. The development of differential equations is linked to the general development of mathematics. The subject of differential equations originated in the study of calculus. The fundamental ideas of the Calculus: the processes of differentiation and integration and the fundamental theorem which relates them were independently developed by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the seventeenth century. In Newton’s case, a primary motivation for the development was to provide a tool for solving problems involving motion and other physical phenomena. In the first uses of differentiation and integration to solve problems, the term differential equations had not yet been formalized. Rather, the investigators used derivatives and differentials without distinction and derived particular equations for particular problems, which they attempted to solve by any method that occurred to them. About the beginning of the eighteenth century, several categories of problems dominated scientific investigations and led to the consideration of differential equations and methods for their solution. This tool since then has being used to describe, understand and predict behavior of many physical processes or system through the construction of mathematical models of the system in diverse areas of study including physics, economics, epidemiology, fluid dynamics, pharmacology, social sciences and ecology , just to mention a few. Ecology is one of the interesting areas that have been studied using mathematical models. -
Primary and Metastatic Tumor Dormancy As a Result of Population Heterogeneity Irina Kareva
Kareva Biology Direct (2016) 11:37 DOI 10.1186/s13062-016-0139-0 HYPOTHESIS Open Access Primary and metastatic tumor dormancy as a result of population heterogeneity Irina Kareva Abstract Existence of tumor dormancy, or cancer without disease, is supported both by autopsy studies that indicate presence of microscopic tumors in men and women who die of trauma (primary dormancy), and by long periods of latency between excision of primary tumors and diseaserecurrence(metastaticdormancy).Withindormant tumors, two general mechanisms underlying the dynamics are recognized, namely, the population existing at limited carrying capacity (tumor mass dormancy), and solitary cell dormancy, characterized by long periods of quiescence marked by cell cycle arrest. Here we focus on mechanisms that precede the avascular tumor reaching its carrying capacity, and propose that dynamics consistent with tumor dormancy and subsequent escape from it can be accounted for with simple models that take into account population heterogeneity. We evaluate parametrically heterogeneous Malthusian, logistic and Allee growth models and show that 1) time to escape from tumor dormancy is driven by the initial distribution of cell clones in the population and 2) escape from dormancy is accompanied by a large increase in variance, as well as the expected value of fitness-determining parameters. Based on our results, we propose that parametrically heterogeneous logistic model would be most likely to account for primary tumor dormancy, while distributed Allee model would be most appropriate for metastatic dormancy. We conclude with a discussion of dormancy as a stage within a larger context of cancer as a systemic disease. Reviewers: This article was reviewed by Heiko Enderling and Marek Kimmel.