出版社: Belknap Press
副标题: Exploring the Equations of Life
出版年: 2006929
页数: 384
定价: USD 51.50
装帧: hardcover
ISBN: 9780674023383
内容简介 · · · · · ·
At a time of unprecedented expansion in the life sciences, evolution is the one theory that transcends all of biology. Any observation of a living system must ultimately be interpreted in the context of its evolution. Evolutionary change is the consequence of mutation and natural selection, which are two concepts that can be described by mathematical equations.Evolutionary Dyna...
At a time of unprecedented expansion in the life sciences, evolution is the one theory that transcends all of biology. Any observation of a living system must ultimately be interpreted in the context of its evolution. Evolutionary change is the consequence of mutation and natural selection, which are two concepts that can be described by mathematical equations.Evolutionary Dynamics is concerned with these equations of life. In this book, Martin Nowak draws on the languages of biology and mathematics to outline the mathematical principles according to which life evolves. His work introduces readers to the powerful yet simple laws that govern the evolution of living systems, no matter how complicated they might seem.
Evolution has become a mathematical theory, Nowak suggests, and any idea of an evolutionary process or mechanism should be studied in the context of the mathematical equations of evolutionary dynamics. His book presents a range of analytical tools that can be used to this end: fitness landscapes, mutation matrices, genomic sequence space, random drift, quasispecies, replicators, the Prisoner's Dilemma, games in finite and infinite populations, evolutionary graph theory, games on grids, evolutionary kaleidoscopes, fractals, and spatial chaos. Nowak then shows how evolutionary dynamics applies to critical realworld problems, including the progression of viral diseases such as AIDS, the virulence of infectious agents, the unpredictable mutations that lead to cancer, the evolution of altruism, and even the evolution of human language. His book makes a clear and compelling case for understanding every living systemand everything that arises as a consequence of living systemsin terms of evolutionary dynamics.
作者简介 · · · · · ·
Martin Nowak is Professor of Biology and of Mathematics at Harvard University. He is Director of the Program for Evolutionary Dynamics.
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mmm (这世界上有种爱情叫做一叶障目)
The main ingredients of evolutionary dynamics are reproduction, mutation, selection, random drift, and spatial movement. Always keep in mind that the population is the fundamental basis of any evolution. Individuals, genes, or ideas can change over time, but only populations evolve. what is the replicator equation? it seems to be key equation in the whole theory. Chapter 2. Chapter 3. Chap...20121218 11:50
what is the replicator equation? it seems to be key equation in the whole theory.Chapter 2.Chapter 3.Chapter 4:The main ingredients of evolutionary dynamics are reproduction, mutation, selection, random drift, and spatial movement. Always keep in mind that the population is the fundamental basis of any evolution. Individuals, genes, or ideas can change over time, but only populations evolve.
evolutionary game dynamics arise whenever the fitness of an individual is not constant but depends on the relative abundance(=frequency) of others in the population.
回应 20121218 11:50 
Evolutionary Dynamics 02 Genetic Diversity of HIV Topics: fitness landscape and sequence space • bioinformatics • sequence space and fitness landscape • natural distribution of mutants • Mutationselection balance leads to localized adaptation in sequence space • Bioinformatics The evolutionary dynamics of genome size and genome organization is a fascinating topic. Infor... (1回应)
20110206 04:22
Evolutionary Dynamics02 Genetic Diversity of HIVTopics: fitness landscape and sequence space• bioinformatics• sequence space and fitness landscape• natural distribution of mutants• Mutationselection balance leads to localized adaptation in sequence space• BioinformaticsThe evolutionary dynamics of genome size and genome organization is a fascinating topic. Information stored in onedimensional sequence can recover a fourdimensional organism with a designed plan of development (the timedimension included). Molecular biology adds a precise informationtheoretic perspective to evolutionary dynamics, while theoretical informatics is a bit lagging behind to do data mining.• Sequence space and fitness landscapeEvolution is a trajectory through sequence space. This trajectory needs an efficient guide. There is a mapping from a genotype to phenotype, and another mapping from phenotype to fitness. The fitness landscape is a convolution of these two mappings. A practical definition of fitness, as previously seen, is effective reproduction rate (birth rate – death rate). That is applicable to single organisms such as bacteria. However, it is less applicable to organisms having complex life span, early population biology have attempted to account for the discrepancy by building models involving age structures.• Natural distribution of mutantsQuasispecies equation:If Q is the identify matrix, i.e. the replication is error free, the equation recovers to dxi/dt = xi (fi φ), “survival of the fittest”. If replication is with error, the system converges to a global attractor in the interior of the simplex.The equilibrium solution does not necessarily maximize the average fitness φ.## definition of biological species can be taken as a cluster of quasispecies centered on one sequence.An attempt on definition of q:Assume only point mutation (no indels) on binary sequence of length L, mutation probability is u, thenqij = uhij (1 – u)L  hijwhere hij := Haming distance## An naïve attempt though.Given HIV’s point mutation rate and genome length, probability of generating oneerror sequence is 0.22. Considering the number of newly infected cells per day, oneerror mutant is capable of arise 2.2 * 105 times per day. This number signifies the enormous potential of HIV to escape from selection pressures that are meant to control them.• Mutationselection balance leads to localized adaptation in sequence spaceAdaption is defined as quasispecies is able to find peaks in the fitness landscape and stays there.## note that in usual sense, adaptation is defined as an organism’s ability to change its phenotype so as to adapt to environmental changes. Clearly here, such capability in absorbed in the functional definition of fitness. Because of mutationselection balance, quasispecies tends to ascend and localize on local peak of fitness landscape. There is a condition that too high mutation rate prevents such event to happen – a mutational meltdown.WT := x0 with fitness f0 > 1mutant := x1 with fitness 1Back mutation from mutant x1 to wildtype is ignored:dx0/dt = x0 (f0q  φ)dx1/dt = x1f0 (1  q) + x1  φx1 => dx0/dt = x0 ( f0q  1  x0 (f0 – 1)) => x0* = 0, x0* = (f0q  1)/(f0  1)If f0q < 1, x0* = 0 is stable, the system will converges to zero wildtype so that mutational meltdown occurs. Hence, mutation rate has to be under an error thresholdf0q > 1=> log f0 > L log (1 – u) /approx Lu=> u < log f0 / L=> /approx u < 1/LThe genomic mutation rate, Lu, has to be smaller than 1. That’s true in many natural organisms.Assume two peaks on fitness landscape.If u < threshold u1, quasispecies is clustered on the higher peak with a narrower spread.If u1 < u < u2, quasispecies is clustered on the lower peak with a broader spread.If u > u2, mutational meltdown.1回应 20110206 04:22 
Evolutionary Dynamics 01 What Is Evolution Topic: forces shaping population dynamics • population growth • limited environmental resource and interspecies competition • species winning based on fitness • sublinear growth rate leads to coexistence • superlinear growth rate leads to winning of species which first occupies ecological niches • mutation leads to coexistence...
20110206 04:12
Evolutionary Dynamics01 What Is EvolutionTopic: forces shaping population dynamics• population growth• limited environmental resource and interspecies competition• species winning based on fitness• sublinear growth rate leads to coexistence• superlinear growth rate leads to winning of species which first occupies ecological niches• mutation leads to coexistence• random mating does not affect allele frequencyThis chapter summarizes result of early mathematical models of population biology.• Population growthDifference equationxt = x0 2tDifferential equationx(t) = x0 ertEven under the same scheme of 20minute time step, the two equations produce different numerical result. That is because of the underlying assumptions adopted here. The difference equation assumes synchronized cell division, while the differential equation assumes asynchronized cell division, as cell division rate follows a probabilistic exponential distribution. Introduce cell deathdx/dt = (r  d) xr/d is taken as basic reproductive ratio. This parameter determines future of the population, i.e. infinite or zero.• Limited environmental resourceLogistic differential equationdx/dt = r x (1  x/K)Stable attractor at x*=K.Logistic difference equationxt+1 = a xt (1xt)analog as a := 1+r, x rescaled to represent fractionThis equation produces deterministic chaos for 3.6786<a≤4.## this equation is an analog to the logistic differential equation.• Interspecies competitionIndependent speciesdx/dt = a xdy/dt = b yset ρ = x/y=> dρ/dt = (ab) ρ=> ρ(t) = ρ0 e(ab)tDominant species is determined only by fitness.Limited by intercompetitioni.e. population size is held const by x+y := 1dx/dt = x (a  φ)dy/dt = y (b  φ)where φ = ax+by = (ax+by)/(x+y) := average fitness of all=> dx/dt = x (1  x) (a – b)Species fitness is reduced by intercompetition, which is represented by average total fitness.Two equilibrium points x*=0, x*=1. Fitness controls which species converges to 1, i.e. “survival of the fitter”.Extend to multi species modelpopulation size is again held constx = (x1, …, xn)φ = ∑ xifi := average fitnessdxi/dt = xi (fi  φ) There will be only one species survive, the one has the largest fitness, i.e. “survival of the fittest.”In trajectory on the simplex, eventually the population will converge to the single global attractor at a corner point.## simplex is cute! • Nonlinear growth leads to coexistence and early control of ecological nichesSet the growth rate to be nonlineardx/dt = a xc  φ xdy/dt = b yc  φ ywhere φ = a xc + b yc=> dx/dt = x (1  x) f(x)where f(x) = a xc1 – b (1x)c1=> x* = 1/(1+(a/b)c1), x* = 0, x* = 1,If c = 1, we are back to the previous linear condition.If c < 1, the growth is subexponential. Coexistence occurs at internal fixed point.If c > 1, the growth is superexponential. Boundary fixed points are stable. Therefore, the attractor to which the system converges to depends on the initial population, if x > x*, the system converges to x > 1, if x < x*, the systems converges to x > 0. The asymptotic behavior is independent of fitness, but which species takes over the ecological niches first, i.e. “survival of the first”. Invasion cannot occur in this case. Invasion is defined as follows: an infinitesimally small fraction of A emerges in all B population, if A can expand and take over B, A is able to invade B. If c > 1, e.g. c = 2, reproduction can occur only if two A’s meet with each other.• Mutation leads to coexistencetwo species modeldx/dt = x (1 – u1) + y u2  φ xdy/dt = y (1 – u2) + x u1  φ yif both species have the same fitness, i.e. a=b=1=> φ = 1=> dx/dt = u2 – x (u1 + u2)=> x* = u2 / (u1 + u2)Mutation leads to coexistence.multi species modelmutation matrix Q = [qij], from i to j<=> The system has a global attractor which is given by the lefthand eigenvector with λ=1.• Mating does not affect allele frequencyAccording to HardyWeinberg principle, in a wellmixed population, random mating does not affect allele frequency. Allele distribution in the population is preserved.回应 20110206 04:12

mmm (这世界上有种爱情叫做一叶障目)
The main ingredients of evolutionary dynamics are reproduction, mutation, selection, random drift, and spatial movement. Always keep in mind that the population is the fundamental basis of any evolution. Individuals, genes, or ideas can change over time, but only populations evolve. what is the replicator equation? it seems to be key equation in the whole theory. Chapter 2. Chapter 3. Chap...20121218 11:50
what is the replicator equation? it seems to be key equation in the whole theory.Chapter 2.Chapter 3.Chapter 4:The main ingredients of evolutionary dynamics are reproduction, mutation, selection, random drift, and spatial movement. Always keep in mind that the population is the fundamental basis of any evolution. Individuals, genes, or ideas can change over time, but only populations evolve.
evolutionary game dynamics arise whenever the fitness of an individual is not constant but depends on the relative abundance(=frequency) of others in the population.
回应 20121218 11:50 
Evolutionary Dynamics 01 What Is Evolution Topic: forces shaping population dynamics • population growth • limited environmental resource and interspecies competition • species winning based on fitness • sublinear growth rate leads to coexistence • superlinear growth rate leads to winning of species which first occupies ecological niches • mutation leads to coexistence...
20110206 04:12
Evolutionary Dynamics01 What Is EvolutionTopic: forces shaping population dynamics• population growth• limited environmental resource and interspecies competition• species winning based on fitness• sublinear growth rate leads to coexistence• superlinear growth rate leads to winning of species which first occupies ecological niches• mutation leads to coexistence• random mating does not affect allele frequencyThis chapter summarizes result of early mathematical models of population biology.• Population growthDifference equationxt = x0 2tDifferential equationx(t) = x0 ertEven under the same scheme of 20minute time step, the two equations produce different numerical result. That is because of the underlying assumptions adopted here. The difference equation assumes synchronized cell division, while the differential equation assumes asynchronized cell division, as cell division rate follows a probabilistic exponential distribution. Introduce cell deathdx/dt = (r  d) xr/d is taken as basic reproductive ratio. This parameter determines future of the population, i.e. infinite or zero.• Limited environmental resourceLogistic differential equationdx/dt = r x (1  x/K)Stable attractor at x*=K.Logistic difference equationxt+1 = a xt (1xt)analog as a := 1+r, x rescaled to represent fractionThis equation produces deterministic chaos for 3.6786<a≤4.## this equation is an analog to the logistic differential equation.• Interspecies competitionIndependent speciesdx/dt = a xdy/dt = b yset ρ = x/y=> dρ/dt = (ab) ρ=> ρ(t) = ρ0 e(ab)tDominant species is determined only by fitness.Limited by intercompetitioni.e. population size is held const by x+y := 1dx/dt = x (a  φ)dy/dt = y (b  φ)where φ = ax+by = (ax+by)/(x+y) := average fitness of all=> dx/dt = x (1  x) (a – b)Species fitness is reduced by intercompetition, which is represented by average total fitness.Two equilibrium points x*=0, x*=1. Fitness controls which species converges to 1, i.e. “survival of the fitter”.Extend to multi species modelpopulation size is again held constx = (x1, …, xn)φ = ∑ xifi := average fitnessdxi/dt = xi (fi  φ) There will be only one species survive, the one has the largest fitness, i.e. “survival of the fittest.”In trajectory on the simplex, eventually the population will converge to the single global attractor at a corner point.## simplex is cute! • Nonlinear growth leads to coexistence and early control of ecological nichesSet the growth rate to be nonlineardx/dt = a xc  φ xdy/dt = b yc  φ ywhere φ = a xc + b yc=> dx/dt = x (1  x) f(x)where f(x) = a xc1 – b (1x)c1=> x* = 1/(1+(a/b)c1), x* = 0, x* = 1,If c = 1, we are back to the previous linear condition.If c < 1, the growth is subexponential. Coexistence occurs at internal fixed point.If c > 1, the growth is superexponential. Boundary fixed points are stable. Therefore, the attractor to which the system converges to depends on the initial population, if x > x*, the system converges to x > 1, if x < x*, the systems converges to x > 0. The asymptotic behavior is independent of fitness, but which species takes over the ecological niches first, i.e. “survival of the first”. Invasion cannot occur in this case. Invasion is defined as follows: an infinitesimally small fraction of A emerges in all B population, if A can expand and take over B, A is able to invade B. If c > 1, e.g. c = 2, reproduction can occur only if two A’s meet with each other.• Mutation leads to coexistencetwo species modeldx/dt = x (1 – u1) + y u2  φ xdy/dt = y (1 – u2) + x u1  φ yif both species have the same fitness, i.e. a=b=1=> φ = 1=> dx/dt = u2 – x (u1 + u2)=> x* = u2 / (u1 + u2)Mutation leads to coexistence.multi species modelmutation matrix Q = [qij], from i to j<=> The system has a global attractor which is given by the lefthand eigenvector with λ=1.• Mating does not affect allele frequencyAccording to HardyWeinberg principle, in a wellmixed population, random mating does not affect allele frequency. Allele distribution in the population is preserved.回应 20110206 04:12 
Evolutionary Dynamics 02 Genetic Diversity of HIV Topics: fitness landscape and sequence space • bioinformatics • sequence space and fitness landscape • natural distribution of mutants • Mutationselection balance leads to localized adaptation in sequence space • Bioinformatics The evolutionary dynamics of genome size and genome organization is a fascinating topic. Infor... (1回应)
20110206 04:22
Evolutionary Dynamics02 Genetic Diversity of HIVTopics: fitness landscape and sequence space• bioinformatics• sequence space and fitness landscape• natural distribution of mutants• Mutationselection balance leads to localized adaptation in sequence space• BioinformaticsThe evolutionary dynamics of genome size and genome organization is a fascinating topic. Information stored in onedimensional sequence can recover a fourdimensional organism with a designed plan of development (the timedimension included). Molecular biology adds a precise informationtheoretic perspective to evolutionary dynamics, while theoretical informatics is a bit lagging behind to do data mining.• Sequence space and fitness landscapeEvolution is a trajectory through sequence space. This trajectory needs an efficient guide. There is a mapping from a genotype to phenotype, and another mapping from phenotype to fitness. The fitness landscape is a convolution of these two mappings. A practical definition of fitness, as previously seen, is effective reproduction rate (birth rate – death rate). That is applicable to single organisms such as bacteria. However, it is less applicable to organisms having complex life span, early population biology have attempted to account for the discrepancy by building models involving age structures.• Natural distribution of mutantsQuasispecies equation:If Q is the identify matrix, i.e. the replication is error free, the equation recovers to dxi/dt = xi (fi φ), “survival of the fittest”. If replication is with error, the system converges to a global attractor in the interior of the simplex.The equilibrium solution does not necessarily maximize the average fitness φ.## definition of biological species can be taken as a cluster of quasispecies centered on one sequence.An attempt on definition of q:Assume only point mutation (no indels) on binary sequence of length L, mutation probability is u, thenqij = uhij (1 – u)L  hijwhere hij := Haming distance## An naïve attempt though.Given HIV’s point mutation rate and genome length, probability of generating oneerror sequence is 0.22. Considering the number of newly infected cells per day, oneerror mutant is capable of arise 2.2 * 105 times per day. This number signifies the enormous potential of HIV to escape from selection pressures that are meant to control them.• Mutationselection balance leads to localized adaptation in sequence spaceAdaption is defined as quasispecies is able to find peaks in the fitness landscape and stays there.## note that in usual sense, adaptation is defined as an organism’s ability to change its phenotype so as to adapt to environmental changes. Clearly here, such capability in absorbed in the functional definition of fitness. Because of mutationselection balance, quasispecies tends to ascend and localize on local peak of fitness landscape. There is a condition that too high mutation rate prevents such event to happen – a mutational meltdown.WT := x0 with fitness f0 > 1mutant := x1 with fitness 1Back mutation from mutant x1 to wildtype is ignored:dx0/dt = x0 (f0q  φ)dx1/dt = x1f0 (1  q) + x1  φx1 => dx0/dt = x0 ( f0q  1  x0 (f0 – 1)) => x0* = 0, x0* = (f0q  1)/(f0  1)If f0q < 1, x0* = 0 is stable, the system will converges to zero wildtype so that mutational meltdown occurs. Hence, mutation rate has to be under an error thresholdf0q > 1=> log f0 > L log (1 – u) /approx Lu=> u < log f0 / L=> /approx u < 1/LThe genomic mutation rate, Lu, has to be smaller than 1. That’s true in many natural organisms.Assume two peaks on fitness landscape.If u < threshold u1, quasispecies is clustered on the higher peak with a narrower spread.If u1 < u < u2, quasispecies is clustered on the lower peak with a broader spread.If u > u2, mutational meltdown.1回应 20110206 04:22

mmm (这世界上有种爱情叫做一叶障目)
The main ingredients of evolutionary dynamics are reproduction, mutation, selection, random drift, and spatial movement. Always keep in mind that the population is the fundamental basis of any evolution. Individuals, genes, or ideas can change over time, but only populations evolve. what is the replicator equation? it seems to be key equation in the whole theory. Chapter 2. Chapter 3. Chap...20121218 11:50
what is the replicator equation? it seems to be key equation in the whole theory.Chapter 2.Chapter 3.Chapter 4:The main ingredients of evolutionary dynamics are reproduction, mutation, selection, random drift, and spatial movement. Always keep in mind that the population is the fundamental basis of any evolution. Individuals, genes, or ideas can change over time, but only populations evolve.
evolutionary game dynamics arise whenever the fitness of an individual is not constant but depends on the relative abundance(=frequency) of others in the population.
回应 20121218 11:50 
Evolutionary Dynamics 02 Genetic Diversity of HIV Topics: fitness landscape and sequence space • bioinformatics • sequence space and fitness landscape • natural distribution of mutants • Mutationselection balance leads to localized adaptation in sequence space • Bioinformatics The evolutionary dynamics of genome size and genome organization is a fascinating topic. Infor... (1回应)
20110206 04:22
Evolutionary Dynamics02 Genetic Diversity of HIVTopics: fitness landscape and sequence space• bioinformatics• sequence space and fitness landscape• natural distribution of mutants• Mutationselection balance leads to localized adaptation in sequence space• BioinformaticsThe evolutionary dynamics of genome size and genome organization is a fascinating topic. Information stored in onedimensional sequence can recover a fourdimensional organism with a designed plan of development (the timedimension included). Molecular biology adds a precise informationtheoretic perspective to evolutionary dynamics, while theoretical informatics is a bit lagging behind to do data mining.• Sequence space and fitness landscapeEvolution is a trajectory through sequence space. This trajectory needs an efficient guide. There is a mapping from a genotype to phenotype, and another mapping from phenotype to fitness. The fitness landscape is a convolution of these two mappings. A practical definition of fitness, as previously seen, is effective reproduction rate (birth rate – death rate). That is applicable to single organisms such as bacteria. However, it is less applicable to organisms having complex life span, early population biology have attempted to account for the discrepancy by building models involving age structures.• Natural distribution of mutantsQuasispecies equation:If Q is the identify matrix, i.e. the replication is error free, the equation recovers to dxi/dt = xi (fi φ), “survival of the fittest”. If replication is with error, the system converges to a global attractor in the interior of the simplex.The equilibrium solution does not necessarily maximize the average fitness φ.## definition of biological species can be taken as a cluster of quasispecies centered on one sequence.An attempt on definition of q:Assume only point mutation (no indels) on binary sequence of length L, mutation probability is u, thenqij = uhij (1 – u)L  hijwhere hij := Haming distance## An naïve attempt though.Given HIV’s point mutation rate and genome length, probability of generating oneerror sequence is 0.22. Considering the number of newly infected cells per day, oneerror mutant is capable of arise 2.2 * 105 times per day. This number signifies the enormous potential of HIV to escape from selection pressures that are meant to control them.• Mutationselection balance leads to localized adaptation in sequence spaceAdaption is defined as quasispecies is able to find peaks in the fitness landscape and stays there.## note that in usual sense, adaptation is defined as an organism’s ability to change its phenotype so as to adapt to environmental changes. Clearly here, such capability in absorbed in the functional definition of fitness. Because of mutationselection balance, quasispecies tends to ascend and localize on local peak of fitness landscape. There is a condition that too high mutation rate prevents such event to happen – a mutational meltdown.WT := x0 with fitness f0 > 1mutant := x1 with fitness 1Back mutation from mutant x1 to wildtype is ignored:dx0/dt = x0 (f0q  φ)dx1/dt = x1f0 (1  q) + x1  φx1 => dx0/dt = x0 ( f0q  1  x0 (f0 – 1)) => x0* = 0, x0* = (f0q  1)/(f0  1)If f0q < 1, x0* = 0 is stable, the system will converges to zero wildtype so that mutational meltdown occurs. Hence, mutation rate has to be under an error thresholdf0q > 1=> log f0 > L log (1 – u) /approx Lu=> u < log f0 / L=> /approx u < 1/LThe genomic mutation rate, Lu, has to be smaller than 1. That’s true in many natural organisms.Assume two peaks on fitness landscape.If u < threshold u1, quasispecies is clustered on the higher peak with a narrower spread.If u1 < u < u2, quasispecies is clustered on the lower peak with a broader spread.If u > u2, mutational meltdown.1回应 20110206 04:22 
Evolutionary Dynamics 01 What Is Evolution Topic: forces shaping population dynamics • population growth • limited environmental resource and interspecies competition • species winning based on fitness • sublinear growth rate leads to coexistence • superlinear growth rate leads to winning of species which first occupies ecological niches • mutation leads to coexistence...
20110206 04:12
Evolutionary Dynamics01 What Is EvolutionTopic: forces shaping population dynamics• population growth• limited environmental resource and interspecies competition• species winning based on fitness• sublinear growth rate leads to coexistence• superlinear growth rate leads to winning of species which first occupies ecological niches• mutation leads to coexistence• random mating does not affect allele frequencyThis chapter summarizes result of early mathematical models of population biology.• Population growthDifference equationxt = x0 2tDifferential equationx(t) = x0 ertEven under the same scheme of 20minute time step, the two equations produce different numerical result. That is because of the underlying assumptions adopted here. The difference equation assumes synchronized cell division, while the differential equation assumes asynchronized cell division, as cell division rate follows a probabilistic exponential distribution. Introduce cell deathdx/dt = (r  d) xr/d is taken as basic reproductive ratio. This parameter determines future of the population, i.e. infinite or zero.• Limited environmental resourceLogistic differential equationdx/dt = r x (1  x/K)Stable attractor at x*=K.Logistic difference equationxt+1 = a xt (1xt)analog as a := 1+r, x rescaled to represent fractionThis equation produces deterministic chaos for 3.6786<a≤4.## this equation is an analog to the logistic differential equation.• Interspecies competitionIndependent speciesdx/dt = a xdy/dt = b yset ρ = x/y=> dρ/dt = (ab) ρ=> ρ(t) = ρ0 e(ab)tDominant species is determined only by fitness.Limited by intercompetitioni.e. population size is held const by x+y := 1dx/dt = x (a  φ)dy/dt = y (b  φ)where φ = ax+by = (ax+by)/(x+y) := average fitness of all=> dx/dt = x (1  x) (a – b)Species fitness is reduced by intercompetition, which is represented by average total fitness.Two equilibrium points x*=0, x*=1. Fitness controls which species converges to 1, i.e. “survival of the fitter”.Extend to multi species modelpopulation size is again held constx = (x1, …, xn)φ = ∑ xifi := average fitnessdxi/dt = xi (fi  φ) There will be only one species survive, the one has the largest fitness, i.e. “survival of the fittest.”In trajectory on the simplex, eventually the population will converge to the single global attractor at a corner point.## simplex is cute! • Nonlinear growth leads to coexistence and early control of ecological nichesSet the growth rate to be nonlineardx/dt = a xc  φ xdy/dt = b yc  φ ywhere φ = a xc + b yc=> dx/dt = x (1  x) f(x)where f(x) = a xc1 – b (1x)c1=> x* = 1/(1+(a/b)c1), x* = 0, x* = 1,If c = 1, we are back to the previous linear condition.If c < 1, the growth is subexponential. Coexistence occurs at internal fixed point.If c > 1, the growth is superexponential. Boundary fixed points are stable. Therefore, the attractor to which the system converges to depends on the initial population, if x > x*, the system converges to x > 1, if x < x*, the systems converges to x > 0. The asymptotic behavior is independent of fitness, but which species takes over the ecological niches first, i.e. “survival of the first”. Invasion cannot occur in this case. Invasion is defined as follows: an infinitesimally small fraction of A emerges in all B population, if A can expand and take over B, A is able to invade B. If c > 1, e.g. c = 2, reproduction can occur only if two A’s meet with each other.• Mutation leads to coexistencetwo species modeldx/dt = x (1 – u1) + y u2  φ xdy/dt = y (1 – u2) + x u1  φ yif both species have the same fitness, i.e. a=b=1=> φ = 1=> dx/dt = u2 – x (u1 + u2)=> x* = u2 / (u1 + u2)Mutation leads to coexistence.multi species modelmutation matrix Q = [qij], from i to j<=> The system has a global attractor which is given by the lefthand eigenvector with λ=1.• Mating does not affect allele frequencyAccording to HardyWeinberg principle, in a wellmixed population, random mating does not affect allele frequency. Allele distribution in the population is preserved.回应 20110206 04:12
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书里的公式，我愿意用“美”来形容
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伟大
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曾关注过他滴研究，他滴老师Karl sigmund开创性滴研究打开了进化论滴新视野……未来滴生物学领域将有更多滴物理人和数学人参与……这又给同学们多了个选择……
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Make the semester!
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实在是太棒了
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伟大
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Make the semester!
0 有用 robin & cabin 20120601
曾关注过他滴研究，他滴老师Karl sigmund开创性滴研究打开了进化论滴新视野……未来滴生物学领域将有更多滴物理人和数学人参与……这又给同学们多了个选择……
0 有用 狗打肉包子 20101215
实在是太棒了
0 有用 lcy 20090716
书里的公式，我愿意用“美”来形容