Slide 1

Chapter 4 Continuous Models Introduction Independent variables are chosen as continuous values Time t Distance x, .. Paradigm for state variables Change over infinite small interval Change rate, rate of change Ordinary differential equation (ODE) First order, second-order, higher orders System of ODEs Partial differential equations Slide 2 Malthuss Model Thomas R. Malthus (1766-1834): Father of population model In 1798, ``An Essay on the Principle of population Profoundly impact on evolution theory of Charles Darwin (1809-1882) ``Malthus's observation was that, unchecked by environmental or social constraints, it appeared that human populations doubled every twenty-five years, regardless of the initial population size. Said another way, he posited that populations increased by a fixed proportion over a given period of time and that, absent constraints, this proportion was not affected by the size of the population. Slide 3 Malthuss Model ``By way of example, according to Malthus, if a population of 100 individuals increased to a population 135 individuals over the course of, say, five years, then a population of 1000 individuals would increase to 1350 individuals over the same period of time. Let t: time N(t): the number of population at time t Balance equation: Slide 4 Malthuss model Consider the time interval Slide 5 Malthuss model Malthuss assumption: Unlimited resource & no migration Birth rate and death rate are both constants The equation Slide 6 Malthuss model Phenomena Population `explosion: ``story of Prof. Yanchu Ma Population distinction No change World population Slide 7 The logistic model Assumption: (Verhulst, 1836) Limited resource, no migration & death rate is constant Birth rate decreases with increasing population Slide 8 The logistic model The solution: Phenomena Population distinction: Equilibrium: Carrying capacity K: N(t) simply increase monotonically to K Form a sigmoid character: slow-fast-slow change fast-slow change N(t) decreases monotonically to K Slide 9 Slide 10 Equilibrium & stability Consider autonomous ODE Equilibrium: Asymptotically stable: Conditions: Stable: Unstable: Slide 11 Equilibrium & stability Reasons For Malthuss model: N*=0 Stable Unstable For the logistic model: N*=0 or K Stable: N*=K Unstable: N*=0 Slide 12 Population with Harvesting Some examples Population of Singapore or USA: immigration Fish in a pound Whale in the ocean Big environmental problems Fishing grounds collapsed under over-fishing Some animals are in danger of extinction due to indiscriminate hunting Big question: harvesting of renewable resources Optimal harvest & without ruining the resource!!!!! Slide 13 Harvest logistic model (I) Assumption: For fish, plants, etc Limited resource & death rate is constant Harvest depends linearly on the population Birth rate decreases with increasing population Slide 14 Harvest logistic model (I) We can find the solution, but here we are NOT so much interested in the value of N at a specific time instant t!! We are rather interested in The terminal value of N when t goes to infinity?? ecologists who guard against extinction of animal or botanical species Scientists in agriculture who have to control pests Scientists in calculating fishing quotas, determine E!!! Whether the population will die out in a finite period of time?? Will N tend to a limit value when t goes to infinity?? What is the optimal harvest strategy: Almost optimal harvest & the population can self renewable Slide 15 Growth of f(N) Slide 16 Harvest logistic model (I) The solution Phenomena Equilibrium: Yield or harvest is: Maximum harvest: Slide 17 Harvest logistic model (I) Time scale of recovery after harvesting No harvest recovery time: With harvest & 0F Slide 28 Harvest logistic model (II) Case 2: E