Computational Biology, Part 14 Models of Population Dynamics and Population Genetics using Recursion RelationsIllustration from Population Dynamics (after Segel)A recursion relationThe simplest caseSlide 5Slide 6Slide 7A better estimateChanging variablesInteractive demonstration - ExcelParameter lineSlide 12Finding steady statesSteady-state solutionsSlide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Interactive demonstration - MatlabSlide 23Transition valuesSummary: Parameter LineIllustration from Population Genetics (after Segel)Illustration from Population GeneticsSlide 28Slide 29Slide 30Computational Biology, Part 14Models of Population Dynamics and Population Genetics using Recursion RelationsComputational Biology, Part 14Models of Population Dynamics and Population Genetics using Recursion RelationsRobert F. MurphyRobert F. MurphyCopyright Copyright  1996, 1999-2009. 1996, 1999-2009.All rights reserved.All rights reserved.Illustration from Population Dynamics (after Segel)Illustration from Population Dynamics (after Segel)Consider a species of insect that hatches in Consider a species of insect that hatches in the spring, lays eggs in the fall and dies in the spring, lays eggs in the fall and dies in the winter.the winter.Let Let NNii be the number of insects in year be the number of insects in year ii..It is safe to say that the number of insects in It is safe to say that the number of insects in a generation will be a function of the a generation will be a function of the number in the previous generation, that is, number in the previous generation, that is, NNi+1i+1 = = ff(N(Nii))..A recursion relationA recursion relationThis type of equation, This type of equation, NNi+1i+1 = = ff(N(Nii)), is called , is called a a recursion relationrecursion relation. Given . Given NN00 and and ff we we can determine all values of can determine all values of NN..In the simplest case, consider the birthrate In the simplest case, consider the birthrate to be constant. Let to be constant. Let RR be the ratio of the be the ratio of the birthrate in the next generation to the birthrate in the next generation to the birthrate in the current generation.birthrate in the current generation.The simplest caseThe simplest caseThen Then ff(N(Nii)) = = RRNNii..Quite naturally, the behavior of Quite naturally, the behavior of NNii depends depends on on RR..RR<1 <1 NNii  ??The simplest caseThe simplest caseThen Then ff(N(Nii)) = = RRNNii..Quite naturally, the behavior of Quite naturally, the behavior of NNii depends depends on on RR..RR<1 <1 NNii  00RR=1 =1  NNii  ??The simplest caseThe simplest caseThen Then ff(N(Nii)) = = RRNNii..Quite naturally, the behavior of Quite naturally, the behavior of NNii depends depends on on RR..RR<1 <1 NNii  00RR=1 =1  NNii  NN00RR>1 >1  NNii  ??The simplest caseThe simplest caseThen Then ff(N(Nii)) = = RRNNii..Quite naturally, the behavior of Quite naturally, the behavior of NNii depends depends on on RR..RR<1 <1 NNii  00RR=1 =1  NNii  NN00RR>1 >1  NNii  A better estimateA better estimateUnlimited growth is unrealistic; eventually Unlimited growth is unrealistic; eventually something (e.g., food supply) will limit something (e.g., food supply) will limit growth.growth.Assume Assume RR changes with changes with NNii. Assume it . Assume it decreases linearly as decreases linearly as NNii increases increasesRR((NNii) = ) = rr[1-[1-NNii//KK] with ] with r,K r,K > 0> 0ThenThenNNi+1i+1 = = rrNNii[1-[1-NNii//KK]]Changing variablesChanging variablesNNi+1i+1 = = rrNNii[1-[1-NNii//KK] implies that when] implies that when N Nii==KK, , the birthrate is zero. When the birthrate is zero. When NNii>>KK, the , the birthrate is negative. Since a negative birthrate is negative. Since a negative birthrate is meaningless, we can only birthrate is meaningless, we can only interpret results when interpret results when NNii<=<=KK..To simplify, let To simplify, let xxii==NNii//KK. Then. Thenxxi+1i+1 = = rr xxi i (1-(1-xxii))Interactive demonstration - ExcelInteractive demonstration - Excel(Demonstration D6)(Demonstration D6)Parameter lineParameter lineFrom our modeling, we conclude that the From our modeling, we conclude that the system shows system shows qualitative lyqualitative ly different different behavior for different parameter (r) values.behavior for different parameter (r) values.We can construct a We can construct a parameter line parameter line to to illustrate this.illustrate this.0 1 2 3 rmonotonic monotonic oscillatoryunstableoscillatoryParameter lineParameter lineWe can construct this line by hand (by We can construct this line by hand (by exploring the behavior of the system using exploring the behavior of the system using the spreadsheet) or we can try to automate the spreadsheet) or we can try to automate the construction of the line.the construction of the line.To do so we can try to find the “final” or To do so we can try to find the “final” or steady statesteady state value of the system for various value of the system for various values of values of rr..Finding steady statesFinding steady statesThe spreadsheet we have looked at uses The spreadsheet we have looked at uses a row for each generation of insects. a row for each generation of insects. We don’t necessarily know how many We don’t necessarily know how many generations are required to reach a generations are required to reach a steady state. One solution is to expand steady state. One solution is to expand the simulation to include “many” cells the simulation to include “many” cells and assume that if there is a steady and assume that if there is a steady state it will be reached.state it will be reached.Steady-state solutionsSteady-state solutionsWe conclude for certain values of r that the final value of We conclude for certain values of r that the final value of xxii seems to vary with seems to vary with rr. What determines the final value?. What determines the final value?We can solve the recursion relation for a steady-state We can solve the recursion relation for a steady-state value. To do so, we look for values of