Lecture 9 - Age and stage structured population models

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Age- and stage-structured population models

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Today’s topics 1 Introduction 2 Matrix Models 3 Reproductive value 4 Stage-structured models

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Motivation Birth and death rates usually depend on age. Introduction Matrix Models Reproductive value Stage-structured models 3 / 28

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Motivation Birth and death rates usually depend on age. Growth rates will diﬀer between populations with diﬀerent age structures. Introduction Matrix Models Reproductive value Stage-structured models 3 / 28

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Motivation Birth and death rates usually depend on age. Growth rates will diﬀer between populations with diﬀerent age structures. Some age classes contribute more to population growth than others. Introduction Matrix Models Reproductive value Stage-structured models 3 / 28

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Motivation Birth and death rates usually depend on age. Growth rates will diﬀer between populations with diﬀerent age structures. Some age classes contribute more to population growth than others. This has important management implications. Introduction Matrix Models Reproductive value Stage-structured models 3 / 28

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Matrix models vs life tables Matrix models • Age is discrete Life tables • Age is continuous Introduction Matrix Models Reproductive value Stage-structured models 4 / 28

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Matrix models vs life tables Matrix models • Age is discrete • Age class is denoted by i Life tables • Age is continuous • Actual age is denoted by x Introduction Matrix Models Reproductive value Stage-structured models 4 / 28

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Matrix models vs life tables Matrix models • Age is discrete • Age class is denoted by i • Each age class can have its own vital rates Life tables • Age is continuous • Actual age is denoted by x • Each age can have its own vital rates Introduction Matrix Models Reproductive value Stage-structured models 4 / 28

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Age-class abundance and age distribution • ni,t is abundance of age class i in year t Introduction Matrix Models Reproductive value Stage-structured models 5 / 28

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Age-class abundance and age distribution • ni,t is abundance of age class i in year t • Suppose initial abundance in the 3 age classes is: n1,0 = 50 (Age class 1, juveniles) n2,0 = 40 (Age class 2, subadults) n3,0 = 10 (Age class 3, adults) • This implies an initial age distribution of: c1,0 = n1,0 /N0 = 0.5 c2,0 = n2,0 /N0 = 0.4 c3,0 = n3,0 /N0 = 0.1 Introduction Matrix Models Reproductive value Stage-structured models 5 / 28

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Age-class abundance and age distribution • ni,t is abundance of age class i in year t • Suppose initial abundance in the 3 age classes is: n1,0 = 50 (Age class 1, juveniles) n2,0 = 40 (Age class 2, subadults) n3,0 = 10 (Age class 3, adults) • This implies an initial age distribution of: c1,0 = n1,0 /N0 = 0.5 c2,0 = n2,0 /N0 = 0.4 c3,0 = n3,0 /N0 = 0.1 An age distribution describes the proportion of individuals in each age class Introduction Matrix Models Reproductive value Stage-structured models 5 / 28

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Age distribution Juveniles Subadults Adults Initial age distribution Proportion in each age class 0.0 0.1 0.2 0.3 0.4 0.5 Introduction Matrix Models Reproductive value Stage-structured models 6 / 28

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Age distribution Declining populations have relatively more old individuals than growing populations Introduction Matrix Models Reproductive value Stage-structured models 7 / 28

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Three age class example We need a model for ni,t+1, abundance of each age class in next time step • Depends on age class survival rates si • And age class birth rates bi Introduction Matrix Models Reproductive value Stage-structured models 8 / 28

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Three age class example We need a model for ni,t+1, abundance of each age class in next time step • Depends on age class survival rates si • And age class birth rates bi n1,t n2,t n3,t s1 s2 b2 × s0 b3 × s0 Introduction Matrix Models Reproductive value Stage-structured models 8 / 28

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Three age class example We need a model for ni,t+1, abundance of each age class in next time step • Depends on age class survival rates si • And age class birth rates bi • Fecundity is often deﬁned as the product of birth rate and oﬀspring survival, fi = bi × s0 n1,t n2,t n3,t s1 s2 f2 f3 Introduction Matrix Models Reproductive value Stage-structured models 8 / 28

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How does this population grow? Age class Equation 1 n1,t+1 = n1,t × f1 + n2,t × f2 + n3,t × f3 Introduction Matrix Models Reproductive value Stage-structured models 9 / 28

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How does this population grow? Age class Equation 1 n1,t+1 = n1,t × f1 + n2,t × f2 + n3,t × f3 2 n2,t+1 = n1,t × s1 Introduction Matrix Models Reproductive value Stage-structured models 9 / 28

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How does this population grow? Age class Equation 1 n1,t+1 = n1,t × f1 + n2,t × f2 + n3,t × f3 2 n2,t+1 = n1,t × s1 3 n3,t+1 = n2,t × s2 Introduction Matrix Models Reproductive value Stage-structured models 9 / 28

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Example Let’s choose values of si and fi Age class Initial population size Survival probability Fecundity rate ni,0 si fi 1 50 0.5 0.0 2 40 0.6 0.8 3 10 0.0 1.7 Introduction Matrix Models Reproductive value Stage-structured models 10 / 28

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Population size, ni,t q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 5 10 15 20 25 30 0 10 20 30 40 50 60 Time Population size q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Age class 1 Age class 2 Age class 3 Introduction Matrix Models Reproductive value Stage-structured models 11 / 28

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Growth rates, ni,t+1 /ni,t = λi,t q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 5 10 15 20 25 30 1.0 1.5 2.0 Time Population growth rate (lambda) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Age class 1 Age class 2 Age class 3 Introduction Matrix Models Reproductive value Stage-structured models 12 / 28

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Age distribution, ci,t q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 5 10 15 20 25 30 0.0 0.2 0.4 0.6 0.8 1.0 Time Proportion in age class q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Age class 1 Age class 2 Age class 3 Introduction Matrix Models Reproductive value Stage-structured models 13 / 28

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Important things to note Age distribution converges to a stable age distribution when survival and fecundity rates are constant. Introduction Matrix Models Reproductive value Stage-structured models 14 / 28

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Important things to note Age distribution converges to a stable age distribution when survival and fecundity rates are constant. Stable age distribution is the proportion of individuals in each age class when the population converges. Growth rates of each age class diﬀer at ﬁrst, but converge once the stable age distribution is reached. Introduction Matrix Models Reproductive value Stage-structured models 14 / 28

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Important things to note Age distribution converges to a stable age distribution when survival and fecundity rates are constant. Stable age distribution is the proportion of individuals in each age class when the population converges. Growth rates of each age class diﬀer at ﬁrst, but converge once the stable age distribution is reached. Asymptotic growth rate is λ (without subscript). Introduction Matrix Models Reproductive value Stage-structured models 14 / 28

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Important things to note Age distribution converges to a stable age distribution when survival and fecundity rates are constant. Stable age distribution is the proportion of individuals in each age class when the population converges. Growth rates of each age class diﬀer at ﬁrst, but converge once the stable age distribution is reached. Asymptotic growth rate is λ (without subscript). Growth rate at the stable age distribution is the same for all age classes, and it is geometric! Introduction Matrix Models Reproductive value Stage-structured models 14 / 28

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Matrix models Matrix models aren’t actually “new” models. Introduction Matrix Models Reproductive value Stage-structured models 15 / 28

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Matrix models Matrix models aren’t actually “new” models. They are the same old models we have been talking about. Introduction Matrix Models Reproductive value Stage-structured models 15 / 28

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Matrix models Matrix models aren’t actually “new” models. They are the same old models we have been talking about. But, they make it much easier to compute important quantities like λ and reproductive value. Introduction Matrix Models Reproductive value Stage-structured models 15 / 28

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What is a matrix? Deﬁnition: A matrix is a rectangular array of numbers • Usually denoted by an uppercase, bold letter • Either square or rounded brackets are used • Usually, rows are indexed by i, columns by j Introduction Matrix Models Reproductive value Stage-structured models 16 / 28

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What is a matrix? Deﬁnition: A matrix is a rectangular array of numbers • Usually denoted by an uppercase, bold letter • Either square or rounded brackets are used • Usually, rows are indexed by i, columns by j Example of a 3 × 4 matrix: A =   a1,1 a1,2 a1,3 a1,4 a2,1 a2,2 a2,3 a2,4 a3,1 a3,2 a3,3 a3,4   Introduction Matrix Models Reproductive value Stage-structured models 16 / 28

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Leslie matrix What is it? • Square matrix • Fertilities on ﬁrst row • Survival probs on lower oﬀ-diagonal Introduction Matrix Models Reproductive value Stage-structured models 17 / 28

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Leslie matrix What is it? • Square matrix • Fertilities on ﬁrst row • Survival probs on lower oﬀ-diagonal Example: A =     f1 f2 f3 f4 s1 0 0 0 0 s2 0 0 0 0 s3 0     Introduction Matrix Models Reproductive value Stage-structured models 17 / 28

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How do we use a Leslie matrix? These two expressions are equivalent: Age class Equation 1 n1,t+1 = n1,t × f1 + n2,t × f2 + n3,t × f3 2 n2,t+1 = n1,t × s1 3 n3,t+1 = n2,t × s2 AND nt+1 = A × nt Introduction Matrix Models Reproductive value Stage-structured models 18 / 28

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Matrix multiplication =     a b c d e f g h i j k l m n o p     ×     w x y z     Introduction Matrix Models Reproductive value Stage-structured models 19 / 28

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Matrix multiplication     aw + bx + cy + dz ew + fx + gy + hz iw + jx + ky + lz mw + nx + oy + pz     =     a b c d e f g h i j k l m n o p     ×     w x y z     Introduction Matrix Models Reproductive value Stage-structured models 19 / 28

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Matrix multiplication and Leslie matrix     n1,t+1 n2,t+1 n3,t+1 n4,t+1     =     f1 f2 f3 f4 s1 0 0 0 0 s2 0 0 0 0 s3 0     ×     n1,t n2,t n3,t n4,t     Introduction Matrix Models Reproductive value Stage-structured models 20 / 28

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Reproductive Value Deﬁnition: The extent to which an individual in age class i will contribute to the ancestry of future generations. Introduction Matrix Models Reproductive value Stage-structured models 21 / 28

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Reproductive Value Deﬁnition: The extent to which an individual in age class i will contribute to the ancestry of future generations. vi = I j=i j−1 h=i sh fj λi−j−1 Introduction Matrix Models Reproductive value Stage-structured models 21 / 28

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Reproductive Value Deﬁnition: The extent to which an individual in age class i will contribute to the ancestry of future generations. vi = I j=i j−1 h=i sh fj λi−j−1 Fact: A post-reproductive individual will have a reproductive value of zero Introduction Matrix Models Reproductive value Stage-structured models 21 / 28

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Reproductive Value Deﬁnition: The extent to which an individual in age class i will contribute to the ancestry of future generations. vi = I j=i j−1 h=i sh fj λi−j−1 Fact: A post-reproductive individual will have a reproductive value of zero Question: Will a ﬁrst-year individual have a higher or lower reproductive value than a second-year individual? Introduction Matrix Models Reproductive value Stage-structured models 21 / 28

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Reproductive value for previous example Age class 1 Age class 2 Age class 3 Fisher's reproductive value 0.0 0.5 1.0 1.5 2.0 Introduction Matrix Models Reproductive value Stage-structured models 22 / 28

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Reproductive value Sir Ronald Aylmer Fisher was central to the development of the idea of reproductive value. Introduction Matrix Models Reproductive value Stage-structured models 23 / 28

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Other important quantities Net reproductive rate The expected number of individuals produced by an individual over its lifetime R0 = I i=1 fi i−1 j=1 sj Introduction Matrix Models Reproductive value Stage-structured models 24 / 28

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Other important quantities Net reproductive rate The expected number of individuals produced by an individual over its lifetime R0 = I i=1 fi i−1 j=1 sj Generation time The time required for a population to increase by a factor of R0 T = log(R0) log(λ) Introduction Matrix Models Reproductive value Stage-structured models 24 / 28

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Other properties of the Leslie matrix1 The dominant eigenvalue of A is the growth rate λ. 1This is for graduate students only Introduction Matrix Models Reproductive value Stage-structured models 25 / 28

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Other properties of the Leslie matrix1 The dominant eigenvalue of A is the growth rate λ. The right eigenvector is the stable age distribution. 1This is for graduate students only Introduction Matrix Models Reproductive value Stage-structured models 25 / 28

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Other properties of the Leslie matrix1 The dominant eigenvalue of A is the growth rate λ. The right eigenvector is the stable age distribution. The left eigenvector is Fisher’s reproductive value. 1This is for graduate students only Introduction Matrix Models Reproductive value Stage-structured models 25 / 28

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Stage-structured population models • Age isn’t always the best way to think about population structure Introduction Matrix Models Reproductive value Stage-structured models 26 / 28

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Stage-structured population models • Age isn’t always the best way to think about population structure • For some populations, it is much more useful to think about size structure or even spatial structure. • These “stage-structured” models diﬀer from age-structured models in that individuals can remain in a stage class (with probability 1 − pi) for multiple time periods. Introduction Matrix Models Reproductive value Stage-structured models 26 / 28

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Stage-structured population models • Age isn’t always the best way to think about population structure • For some populations, it is much more useful to think about size structure or even spatial structure. • These “stage-structured” models diﬀer from age-structured models in that individuals can remain in a stage class (with probability 1 − pi) for multiple time periods. n1,t n2,t n3,t s1 p2 s2 f2 f3 (1 − p2 )s2 s3 Introduction Matrix Models Reproductive value Stage-structured models 26 / 28

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Stage-structured population models In stage-structured models, individuals transition from one stage to the next with probability pi . Introduction Matrix Models Reproductive value Stage-structured models 27 / 28

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Stage-structured population models In stage-structured models, individuals transition from one stage to the next with probability pi . We can add these transition probabilities to our projection matrix like this: Introduction Matrix Models Reproductive value Stage-structured models 27 / 28

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Stage-structured population models In stage-structured models, individuals transition from one stage to the next with probability pi . We can add these transition probabilities to our projection matrix like this: A =     f1 f2 f3 f4 s1 s2 (1 − p2 ) 0 0 0 s2 p2 s3 (1 − p3 ) 0 0 0 s3 p3 s4     Introduction Matrix Models Reproductive value Stage-structured models 27 / 28

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Summary Vital rates (s and f) are usually age-speciﬁc. Introduction Matrix Models Reproductive value Stage-structured models 28 / 28

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Summary Vital rates (s and f) are usually age-speciﬁc. Population growth will depend on age distribution. Introduction Matrix Models Reproductive value Stage-structured models 28 / 28

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Summary Vital rates (s and f) are usually age-speciﬁc. Population growth will depend on age distribution. If vital rates are constant, population will reach stable age distribution with constant growth rate λ. Reproductive value indicates which age class contributes the most to population growth. Introduction Matrix Models Reproductive value Stage-structured models 28 / 28

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Summary Vital rates (s and f) are usually age-speciﬁc. Population growth will depend on age distribution. If vital rates are constant, population will reach stable age distribution with constant growth rate λ. Reproductive value indicates which age class contributes the most to population growth. Matrix models are a convenient method used to work with age-structured populations. Introduction Matrix Models Reproductive value Stage-structured models 28 / 28