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A Introduction to Modeling Population Dynamics Bridging the gap between education and technology Why Model Populations? • The size of the human population impacts our lives as both citizens of the world and members of our local communities. – Will the population of my town, region, or state grow or decrease and what will be the economic impact of those changes? – Will the world population grow to a point that is not sustainable? Bridging the gap between education and technology. Why Model Populations • The size variations of non-human populations have relevance to us. – Will our once-abundant fish species exist in sufficient numbers to be harvested for food? – Will populations of endangered animals continue to be affected by human interventions? • For more population research topics, check forest.bio.ic.ac.uk/cpb/cpb/programme.html Bridging the gap between education and technology. Presentation Overview • Illustrate the development of some basic one- and two-species population models. • Present four models of population growth: – Malthusian (exponential)growth – Logistics growth – Logistics growth with harvesting – Predator-Prey interaction Bridging the gap between education and technology. The Malthus Model of Population Growth • Proposed in 1798 by the Englishman, Thomas R. Malthus. • By observing human populations, he conjectured that Populations appeared to increase by a fixed proportion over a given period of time, and that, in the absence of constraints, this proportion is not affected by the size of the population. Bridging the gap between education and technology. Motivation for Malthus’ Model Graphs of US population available to Malthus. Bridging the gap between education and technology. The Malthus Model To develop a mathematical representation of Malthus’ model, we use the following hypothesis as our governing principle. Populations appeared to increase by a fixed proportion over a given period of time, and that, in the absence of constraints, this proportion is not affected by the size of the population. Bridging the gap between education and technology. The Malthus Model Notation: – t0, t1, t2, …, tN: discrete times at which the population is determined; all ti are equally spaced with time step h = ti+1 - ti for all i. – P0, P1, P2, …, PN: populations at times t0, t1, t2, …, tN, respectively. – b and d: birth and death rates, respectively (with units consistent with those used to measure time); – c = b – d: effective growth rate. Bridging the gap between education and technology. The Malthus Model Mathematical Equation: (Pi + 1 - Pi) / Pi = c*h or Pi + 1 = Pi + c*h*Pi for i = 0, 1, …, N. The initial population, P0, is given at the initial time, t0. Bridging the gap between education and technology. An Example Example: Let t0 = 1900, P0 = 76.1 million (US population in 1900) and c = 0.01 (a per capita growth rate of 1.0% per year). Determine the population at the end of 1, 2, and 3 years, assuming the time step h = 1 Bridging the gap between education and technology. Example Calculation P0 = 76.1; t0 = 1900; h = 1; c = 0.01 P1 = P0 + c*h*P0 = 76.1 + 0.01*1*76.1 = 76.9; t1 = t0 + h = 1900 + 1 = 1901 P2 = P1 + c*h*P1 = 76.9 + 0.01*1*76.9 = 77.7; t2 = 1901 + 1 = 1902 P3 = P2 + c*h*P2 = 77.7 + 0.01*1*77.7 = 78.5; t3 = 1902 + 1 = 1903 Bridging the gap between education and technology. Pseudo Code INPUT: T0 – initial time P0 – population at t0 H – length of time interval N – number of time steps C – population growth rate Bridging the gap between education and technology. Pseudo Code OUTPUT Ti – ith time value Pi – population at ti for i = 0, 1, …, N ALGORITHM: Set T = T0 Set P = P0 Print T, P Bridging the gap between education and technology. Pseudo Code For i = 1, 2, …, N Set T = T + i*h Set P = P + C*H*P Print T, P End For Bridging the gap between education and technology. Malthusian Curves Malthus model plots: c > 0. Bridging the gap between education and technology. Malthusian Curves Malthus model plots: c < 0. Bridging the gap between education and technology. A Malthusian Population Prediction • Malthus model prediction of the US population for the period 1750 – 1998, with initial data taken in 1750: t0 = 1750, P0 = 1,000,000, c = 0.025 Prediction is plotted against actual US population for period 1900-1998. Malthus Plot Bridging the gap between education and technology. Logistic Growth • Results from Malthus model are not valid over long time periods for humans or for other biological forms. • A new model was proposed in 1838 by Belgium mathematician Pierre-Francois Vehulst: growth rate depended on population size. • Result was rediscovered in 1920 by Raymond Pearl and Lowell Reed and used to model US population, as well as populations of fruit flies and yeast cells. Bridging the gap between education and technology. Logistic Growth Pearl-Reed growth rate model in which rate depends on population: b = b0 - b1*P and d = d0 + d1*P Assigning c = b0 - d0, r = b1 + d1, and K = c/r, and substituting new values for b and d into the Malthus equation gives a logistics equation: Bridging the gap between education and technology. Logistic Growth Pi+1 = Pi + c*h *Pi*(1 - Pi/K) • K is the population carrying capacity. – If any Pi = K, equation gives Pi+1 = Pi for all future times. – For Pi = K, birth and death rates are equal. • K has biological meaning for populations with strong interaction among individuals that control their reproduction Bridging the gap between education and technology. Example: Growth of Yeast Cells Population of yeast cells grown under laboratory conditions. Bridging the gap between education and technology. Logistics Growth with Harvesting Harvesting populations, removing members of a population from their environment, is a real-world phenomenon. Assumptions: – Harvesting intensity constant. – Per unit time, each member of the population has chance of being harvested. – In time period h, expected number of harvests is f*h*Pi. where f is harvesting intensity factor. Bridging the gap between education and technology. Logistics Growth with Harvesting The logistic model can easily by modified to include the effect of harvesting: Pi+1 = Pi + c*h*Pi*(1 - Pi/K) - f*h*Pi or Pi+1 = Pi + c*h*Pi*(1 - Pi/K), where c = c - f, K = [(c – f)/c]*K Harvesting Plots Bridging the gap between education and technology. A Predator-Prey Model: two competing fish populations An early predator-prey model • In the mid 1920’s the Italian biologist Umberto D’Ancona was studying the population variations of species of fish that interact with each other. • He came across data on the percentage-of-total- catch of several species of fish that were brought to different Mediterrian ports in the years that spanned World War I Bridging the gap between education and technology. Two competing fish populations Data for the port of Fiume, Italy for the years 1914- 1923: percentage-of-total-catch of predator fish (sharks, skates, rays, etc), not desirable as food fish. Fiume, Italy 40 selachians Percent 30 20 Fiume, Italy 10 0 1910 1915 1920 1925 Years Bridging the gap between education and technology. Two Competing Fish Populations • The level of fishing and its effect on the two fish populations was also of concern to the fishing industry, since it would affect the way fishing was done. • As any good scientist would do, D’Amcona contacted Vito Volterra, a local mathematician, to formulate a model of the growth of predators and their prey and the effect of fishing on the overall population. Bridging the gap between education and technology. Strategy for model development The model development is divided into three stages: 1. In the absence of predators, prey population follows a logistics model and in the absence of prey, predators die out. Predator and prey do not interact with each other and no fishing is allowed. 2. The model is enhanced to allow for predator-prey interaction: predators consume prey 3. Fishing is included in the model Bridging the gap between education and technology. Overall Model Assumptions Simplifications • Only two groups of fish: – prey (food fish) and – predators. • No competing effects among predators • No change in fish populations due to immigration into or emigration out of the physical region occupied by the fish. Bridging the gap between education and technology. Model Development Notation • ti denote specific instances in time • Fi denotes the prey population at time ti • Si denotes the predator population at time ti • cF denotes the growth rate of the prey in the absence of predators • cS denotes the growth rate of the predators in the absence of prey • K denotes the carrying capacity of prey Bridging the gap between education and technology. Stage 1: Basic Model In the absence of predators, the fish population, F, is modeled by Fi+1 = Fi + cF*h*Fi*(1-Fi/K) and in the absence of prey, the predator population, S, is modeled by Si+1 = Si -cS*h*Si Bridging the gap between education and technology. Stage 2: Predator-Prey Interaction a is the prey kill rate due to encounters with predators. Fi+1 = Fi + cF*h*Fi*(1-Fi/K) - a*h*Fi*Si b is a parameter that converts prey-predator encounters to a predator birth rate. Si+1 = Si -cS*h*Si + b*h*Fi*Si Bridging the gap between education and technology. Stage 3: Fishing f is the effective fishing rate for both the predator and prey populations: Fi+1 = Fi + cF*h*Fi*(1-Fi/K) - a*h*Fi*Si - f*h*Fi Si+1 = Si -cS*h*Si + b*h*Fi*Si - f*h*Si Bridging the gap between education and technology. Pseudo Code INPUT: T0 – initial time F0 – prey population at t0 S0 – predator population at t0 H – length of time interval N – number of time steps FR – effective removal rate for fishing CS - growth rate of the prey in the absence of predators CF - growth rate of the predators in the absence of prey Bridging the gap between education and technology. Pseudo Code A – prey kill rate B – predator birth rate K – prey carrying capacity OUTPUT Ti – ith time value Fi – prey population at ti Si – predator population at ti for i = 0, 1, …, N Bridging the gap between education and technology. Pseudo Code ALGORITHM: Set T = t0 Set F = F0 Set S = S0 Print T, F, S For i = 1, 2, …, N Set T = T + i*h Set FTEMP = F Set F = F*(1 + H*(CF*(1 – F/K) – A*S – FR)) Bridging the gap between education and technology. Pseudo Code Set S = S*(1 + H*(B*FTEMP – CS – FR)) Print T, F, S End For Bridging the gap between education and technology. Model Initial Conditions and Parameters Plots for the input values: T0 = 0.0 S0 = 100.0 F0 = 1000.0 H = 0.02 N = 6000.0 FR = 0.005 CS = 0.5 CF = 0.3 A = 0.002 B = 0.0005 K = 4000.0 Predator-Prey Plots Bridging the gap between education and technology. D’Ancona’s Question Answered (Model Solution) A decrease in fishing, f, during WWI decreased the equilibrium prey population, F, and increased the equilibrium predator population, P. f Prey Predators 0.1 800 150 0.01 617 205 0.001 597 211 Bridging the gap between education and technology. General Questions What are some of the weaknesses in our approach to modeling populations that will likely cause the mathematical models to differ from physical reality? – Mathematical modeling is, at some level, an idealistic approach to reality and it is often difficult, if not impossible, to take into account all of the elements in a model that play a role in the real phenomenon. Bridging the gap between education and technology. General Questions – The key to developing a good mathematical model is to determine the essential parameters that govern the phenomenon and incorporate these into the model. A model with say, 100 variables is of little value. Again, a good knowledge of the phenomenon being studied is essential. – In all of our models, we have, for the most part, assumed that constants and parameters are fixed; however, these quantities generally depend on time and hence to achieve more realism, time dependent parameters must be included in the model. Bridging the gap between education and technology. General Questions • What role could population modeling play in such areas as the spread of an infectious disease, the control an insect population, the management of a fish hatchery? – Control of insects: Many insect populations suffer from large populations shifts, due to diseases caused by micro-parasites (e.g., viruses, bacteria, fungi, protozoa). – These pathogens (micro-organisms or viruses that can cause disease) have been used successfully in a variety of pest control measures, as bio-insecticides, and as longer-term bio-control agents. Bridging the gap between education and technology. General Questions - With the correct model, we can ask under what circumstances does the pathogen regulate the host. If we have a choice of pathogens, we could ask which would be the most suitable as a bio-control agent? Bridging the gap between education and technology.

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posted: | 2/14/2012 |

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