Population viability analysis of plant and animal populations with stochastic integral projection models

1

Appendix A - Components of demographic variance for a stochastic IPM

2

This appendix explains how to calculate the terms involved in demographic variance given by

3

equation (9) in the main text. The rationale is based on earlier work by Vindenes et al. (2011),

4

who demonstrate that demographic variance can be calculated from the individual

5

contribution to total reproductive value, called Wx , given that Bx

Wx = S v(Ysx) + ∑v(Ybxi ) , x 

6

Survival

(A1)

i = 1   Reproduction

7

where v(x) is the reproductive value of an individual of size x and other terms are defined

8

in Table A1 below.

9 10

Table A1. Definitions of random variables in equation (A1). Random variable

Distribution law

Description

Sx

Bernoulli(s ( x, z ) )

Bernoulli random variable describing annual survival of individuals with trait value x in environment z

Bx

b(x,z)=E(B x , z )

Fecundity (number of surviving offspring to the next

σ B2 ( x, z ) = Var(B x , z )

time step) of individuals with trait value x in

2 ( x, z ) = Cov(B x , S x , z ) σ BS

environment z characterised by the mean fecundity, the variance in fecundity and the covariance between fecundity and survival

Ysx

f s ( y , x, z )

Random variable describing trait value to the next time step of individuals with trait value x in environment z

Ybxi

f b ( y , x, z )

Random variable describing the size of surviving offspring to the next time step of individuals with trait value x

11 12

In a given environment z, the demographic stochasticity due to an individual with trait value

13

x can be computed using equation (A1)

14

σ d2 ( x, z ) = Var(Wx )

(A2)

1

Population viability analysis of plant and animal populations with stochastic integral projection models 2 2 2 2 )σS (x,  ( )σvb ( x, z ) + µ ( x,z) + b(x, z  x, z z) + z )σ s(x,  vs  vs  

=

15

.

Covariance survival−fecundity

Fecundity

16

Survival

Offspring traitvalue

Growth

2 2 2 , z)µvb(x,  (x (x, z) + 2σ )σB ( z) z )µ x,  x, z µ BS( vs vb 

All terms in equation (A2) are described in Tables A1 and A2.

17 18

Table A2. Definitions of terms in the decomposition of demographic variance from equation (A2). See

19

Table A1 for definition of random variables. Term

Definition

Value

Description

σ vs2 ( x, z )

Var(v(Ysx ))

∫

Sampling variance of adult reproductive value in

2 σ vb ( x, z )

Var(v(Ybxi ))

∫ v ( y, z) f ( y, x, z)dy − µ

µ vs ( x, z )

E(v(Ysx ))

∫ v( y, z) f ( y, x, z)dy

Sample mean of the adult reproductive value in

σ S2 ( x, z )

Var(S x )

s ( x, z )(1 − s ( x, z ) )

Sampling variance of the Bernoulli variable

µ vb ( x, z )

E(v(Ybxi ))

∫

Sample mean of the juvenile reproductive value

v 2 ( y, z ) f s ( y, x, z )dy − µ vs2 ( x, z )

Ω

2

b

Ω

Ω

2 vb ( x, z )

s

environment z Sampling variance of juvenile reproductive value in environment z

environment z

describing annual survival in environment z

v( y, z ) f b ( y, x, z )dy

Ω

in environment z

20 21

To obtain the expression for demographic variance given by equation (9), one must replace all

22

terms by their mean over all environments. The three first terms in equation (9) can be readily

23

computed from the mean kernel. The two last terms in equation (9) require to compute σ B ( x)

24

and σ BS ( x) (variance of the mean fecundity and covariance of the mean fecundity and mean

25

survival, see Table A1), and depend on the exact definition of the fecundity distribution and

26

details about the life cycle.

2

2

27 28

2

Population viability analysis of plant and animal populations with stochastic integral projection models

29

Appendix B - Diffusion approximation

30

Diffusion approximation of the stochastic IPM

31

A continuous time Wiener process can be used to characterise the stochastic dynamics of a

32

finite, structured populations in variable environments (Engen et al. 2005). The approximation

33

requires calculating the drift E(∆V | V )/∆t and the infinitesimal variance Var(∆V | V )/∆t .

34

For the stochastic IPM, these two terms are given by:

(

)

35

Drift : µ = E(∆Vt +1 | Vt )/∆t = λ − 1 V

36

Infinitesimal variance : σ 2 = Var(∆Vt +1 | Vt )/∆t = Var (λ + E + D − 1)Vt | Vt

(B1)

(

37

= Vt 2 Var (Λ t | Vt ) = Vt 2 (Var (E | Vt ) + Var (D | Vt ))

38

= Vt 2 σ e2 + σ d2 /Vt

39

(

)

)

(B2)

The discrete time stochastic dynamic is then equivalent to a continuous time Wiener process Vt + ∆t − Vt = Vt + µ∆t + σ 2 ∆B ,

40

(B3)

41

where ∆B is a Brownian motion term. With the diffusion approximation, it is more common

42

to work on the natural logarithm scale X = ln(V ) because this stabilises the variance. If we

43

assume ∆t = 1 , we can write

Vt +1 = λVt + ε ,

44

(B4)

45

where ε is a random noise variable characterised by E(ε | V ) = 0 , Var (ε | V ) = σ 2 and

46

E ε 2 | V = Var(ε | V ) + E(ε | V ) = σ 2 . Equation (C4) allows computing an equation for the

47

natural logarithm of the total reproductive value

48

49

50

(

)

2

 ε ∆ ln V = ln λ + ln1 +  λVt

 .  

(B5)

The first and second order expansions of equation (C5) are: ∆ ln V = ln λ +

ε + o(ε ) λVt

(B6)

3

Population viability analysis of plant and animal populations with stochastic integral projection models

51

∆ ln V = ln λ +

ε2 ε − 2 + o(ε 2 ) λVt 2λ Vt 2

(B7)

52 53

By integrating expressions for the random noise variable of equation (C4) into the second

54

order expansion (C7), one calculates the drift of the diffusion process characterising X :

55

µ ( X ) = E(∆X | X )/∆t = E(∆ ln V | V ) ,

56

≈ ln λ +

57

≈ ln λ −

58

≈ ln λ −

(

)

E(ε | V ) E ε 2 | V − , 2 λV 2λ V 2 σ2 2

2λ V 2

,

σ e2 + σ d2 /Vt 2λ

2

.

(B8)

59 60

In the same manner, using the first order expansion (C6), the expression for the infinitesimal

61

variance writes like:

62

σ 2 ( X ) = Var(∆X | X )/∆t = Var(∆ ln V | V )

63

 ε  ≈ Var |V    λV

64

≈

65

≈

σ2 2

λ V2 σ e2 + σ d2 /Vt λ

(B9)

2

66 67

Analytical expression for the cumulative extinction risk

68

When the demographic variance is small relative to environmental variation, the diffusion

69

approximation described by equation (C8) and (C9) summarises into the diffusion

70

approximation for large populations in stochastic environments proposed by Lande and 4

Population viability analysis of plant and animal populations with stochastic integral projection models

σ e2

and σ 2 ≈

σ e2

71

Orzack where µ = ln λ −

72

analytical expression of the distribution of time to extinctions and the cumulative probability

73

of extinction before a certain time. In particular, equation (11) in Lande and Orzack gives the

74

cumulative probability of true extinction conditional on initial total reproductive value X0,

75

76

2λ

2

λ

2

. This diffusion approximation allows for an

 − X 0 − µt   X 0 − µt   X 0   G (t X 0 ) = Φ   + exp − 2 µ σ 2 1 − Φ    t t σ σ      where Φ (x) is the standard normal probability integral.

77

5

(B10)

Population viability analysis of plant and animal populations with stochastic integral projection models

78

Appendix C - Life cycle and model parameters for the three case studies

79 80

Redsepal evening primrose (Oenothera glazioviana)

81

The redsepal evening primrose is a dicotyledonous Rosidae plant species with a

82

hermaphroditic flower and a semelparous life cycle that occurs in scattered locations

83

worldwide. The life cycle of the redsepal evening primrose assumes semelparous

84

reproduction followed by establishment of seeds and first year growth of individual seedlings,

85

then followed by annual survival and growth of the rosette until flowering and reproduction

86

(see Rees and Rose 2002 for a detailed justification). After seedling establishment, the plant

87

produces a rosette that grows as large as 5 to 15 cm diameter until sexual maturity is reached

88

after 2 to 5 years. After sexual maturity, a stem grows from the rosette and reaches within a

89

year a size as high as 150 cm. The flower is produced once during the life on top of this stem

90

and the plant dies after flowering. This population is characterised by a strong structuring due

91

to inter-individual variation in the size of the rosette, which influences annual survival

92

probability, annual flowering probability and fecundity. The semelparous life cycle implies a

93

trade-off between early flowering along a life cycle with little mortality risks and late

94

flowering along a life cycle with prolonged growth of the rosette and higher fecundity. Rees

95

and Rose (2002) studied this life history trade-off using a deterministic IPM by choosing an

96

estimate of the seeds’ probability of establishment allowing for the same population growth

97

than the one observed in the field ( pe = 0.0098 and λ ≈ 1.05 ).

98 99

The kernel of this IPM structured by the size of the rosettes of plants is given by

k ( y, x) = p f ( x) f ( x) pe f b ( y, x, Z ) + s a ( x)(1 − p f ( x)) f s ( y, x)   b ( x,Z )

   s( x)

(C1)

100

Parameters values for the terms in this kernel are provided in Table C1a. We found several

101

small differences between the parameter values reported by Kachi and Hirose (1983) and

102

those of the IPM developed by Rees and Rose (2002); thus, we always used parameter values 6

Population viability analysis of plant and animal populations with stochastic integral projection models

103

from the field study. To maintain consistency with previous study (Rees and Rose 2002), we

104

structured the population according to the natural logarithm of rosette size. No estimate of the

105

environmental variance was provided and estimate of the sampling errors were missing for

106

most parameters. To allow a more systematic analysis of differences in population growth rate

107

and demographic variance, variants of the model were derived by changing the seed

108

establishment probability ( pe parameter in Table C1a, tested values ranging from -45% to

109

+15% of the empirical value), and the residual variance of the growth function (tested values

110

ranging from -50% to initial level of the empirical value).

111 112

Table C1a: Model parameters after natural logarithm transformation of rosette size for Oenothera

113

glazioviana obtained from Kachi and Hirose (1983). Trait

Scale

Estimate

SE

Distribution

Binomial distribution

Flowering probability - p f (x) Intercept

Logit

-18.27

NA

Slope rosette size x

Logit

6.91

NA

Fecundity (number of seeds) - f (x) Intercept

log

1.04

NA

Poisson distribution

Slope rosette size x

log

2.22

NA

-4.61

NA

Binomial distribution

0.078

0.573

Gaussian distribution

Binomial distribution

Seed establishment probability - pe Mean value

Logit

Rosette size at one year old - f b ( y, x) Mean value

Identity

Annual survival of rosettes - s a (x) Intercept

Logit

0.36

NA

Slope rosette size x

Logit

0.17

NA

Rosette size at time

114

t +1

- f s ( y, x)

Intercept

Identity

0.96

NA

Slope rosette size x

Identity

0.59

NA

Residual s.d.

Identity

0.67

NA

Notations: SE = standard error of the estimate

115

7

Gaussian distribution

Population viability analysis of plant and animal populations with stochastic integral projection models

116

Table C1b: Effects of changes in model parameters on deterministic growth rate of the mean kernel

117

and demographic variance. Value

λ

σ d2

Value

λ

σ d2

Seed establishment probability - pe

Residual s.d. of the growth function

0.0055

0.91603

1.2965

0.1089

0.8662

0.93344

0.006

0.93459

1.4039

0.1444

0.89341

1.1139

0.0065

0.95226

1.5123

0.1849

0.92103

1.3032

0.007

0.96915

1.6216

0.2304

0.94881

1.498

0.0075

0.98553

1.7317

0.2809

0.97661

1.6964

0.008

1.0009

1.8427

0.3364

1.0044

1.8971

0.0085

1.0159

1.9545

0.3969

1.032

2.099

0.009

1.0354

2.0671

0.4624

1.0595

2.3015

0.0095

1.0444

2.1804

0.5329

1.0869

2.5041

0.01

1.058

2.2944

0.6084

1.1141

2.7059

0.0105

1.0712

2.4092

0.6889

1.1411

2.9064

0.011

1.084

2.5246

0.0115

1.0965

2.6407

118 119

Common lizard (Zootoca vivipara)

120

The common lizard is a ground-dwelling lacertid species from cold and wet environments

121

widely distributed across Eurasia and characterised by a short life cycle (maturation at one to

122

two years old), an indeterminate growth and annual ovoviviparous reproduction. The species

123

is currently not endangered in most of its range but some lowland and southern populations

124

are threatened by global warming (Massot et al. 2008). We used field data collected in one

125

study population monitored since 1989 by Manuel Massot, Jean Clobert and collaborators in

126

the Mont Lozère, Southern France, and recently analysed for temporal variation in life history

127

traits (Le Galliard et al. 2010). Previous investigations of this species have used various forms

128

of density-dependent and density-independent MPM including age and sex-structuring of the

129

population (Le Galliard et al. 2005; Le Galliard et al. 2010; Mugabo et al. 2013); yet, a

130

substantial part of demographic variability can be explained by body size variation among

8

Population viability analysis of plant and animal populations with stochastic integral projection models

131

individuals making the IPM an appropriate framework (González-Suárez et al. 2011).

132

The life cycle of the common lizard counts individuals from the age of one year and

133

assumes annual breeding followed by annual survival and body growth of individual

134

offspring, sub-adults and adults (longevity up to 6-8 years). We assumed that the breeding

135

probability depends on body size, measured by snout vent length (mm), and we calculated the

136

total clutch size (number of eggs), clutch success (number of viable eggs inside the clutch)

137

and female sex ratio (proportion of female offspring among the viable eggs) to estimate the

138

number of female offspring per breeding female per year. The clutch size (average of 5-6

139

offspring) and juvenile’s body size (mean of 38 mm at age 1) are positively correlated to

140

mother’s body size (ranging from 50 to 80 mm). There is also a significant inter-annual

141

variability of body size at birth, annual survival, growth and reproduction. In the population,

142

the total number of females was around 200 individuals in the early 90s. The description of

143

this life cycle is based on data reported in Le Galliard et al. (2010) and reanalysed for the sake

144

of this study.

145 146

The kernel of the IPM structured by the body size of individuals is thus given by:

k ( y, x, Z ) = pb ( x, Z ) f ( x)cs( Z ) p f s j f b ( y, x, Z ) + sa f s ( y, x, Z )    b ( x ,Z )

(C2)

s ( x ,Z )

147

Parameters values for the terms in this kernel are provided in Table D2. Using the field data

148

imported into the statistical software R (R Development Core Team 2005), we run generalised

149

linear models (glm and glmmPQL procedure in R) to estimate breeding probability, litter

150

success and female sex ratio, and linear mixed effects model to estimate body size

151

distributions using the lme procedure in R. We also estimated the total clutch size using a

152

vector generalised linear model (vglm procedure in R) applied on the “zero-truncated” data for

153

clutch size and assumed a generalised Poisson distribution (Kendall and Wittmann 2010).

154

This distribution has two parameters λ and θ , and an upper bound when λ < 0 ; it fitted

155

better the data than the standard Poisson distribution. Annual survival was estimated for 9

Population viability analysis of plant and animal populations with stochastic integral projection models

156

juvenile females using data from offspring born in the laboratory and recaptured until the age

157

of two years old (16 birth cohorts, 1914 individuals). Pooled annual survival was estimated

158

for yearling and adult females using data from all females born in the laboratory or captured

159

for the first time in the field assuming no age-difference in survival (13 years, 784

160

individuals). All estimates of survival were obtained in MARK version 5.1 after correcting for

161

heterogeneous recapture probabilities. Standard errors and inter-annual variation were

162

calculated with the variance component approach that allows to separate sampling from

163

process variation (White and Burnham 1999). For more details on the survival analysis,

164

please refer to Le Galliard et al. (2010).

165 166

Table C2: Parameters in the IPM structured by snout-vent length (SVL, mm) for Zootoca vivipara Trait

Scale

Estimate

SE

Inter-annual SD

Procedure

Distribution

glm

Binomial

vglm

Generalised

Breeding probability - pb ( x, Z ) Intercept

Logit

-29.923

2.618

1.272

Slope SVL x

Logit

0.590

0.050

0

Total clutch size - f (x)

λ

- Mean value

elogit

-1.414

0.092

NA

θ

- Intercept

log

-2.086

0.123

NA

θ

- Slope SVL x

log

0.063

0.00186

NA

Logit

2.640

0.193

0.767

glmmPQL

Binomial

Logit

-0.088

0.028

0

glm

Binomial

-0.982

0.204

0.681

MARK

Binomial

lme

Gaussian

MARK

Binomial

Poisson

Clutch success - cs(Z ) Mean value Female ratio - p f Mean value

Juvenile survival - s j Mean value

Logit

SVL of offspring at one year old - f b ( y, x, Z ) Intercept

Identity

38.454

0.898

3.075

Residual sd*

Power

0.000397

NA

0

0.113

0.295

Sub-adult and adult survival Mean value

Logit

sa

0.213

10

Population viability analysis of plant and animal populations with stochastic integral projection models

SVL of older lizards at time

t + 1 - f s ( y , x, Z )

Intercept

Identity

47.820

0.909

1.909

Slope SVL x

Identity

0.252

0.015

0

Residual sd

Identity

2.818

NA

lme

Gaussian

167

Notations: SE = standard error of the estimate, SD = standard deviation.

168

* Residual variance was modelled like a power function of predicted values with a power exponent of 2.494.

169

11

Population viability analysis of plant and animal populations with stochastic integral projection models

170

Appendix D - Estimation of elasticities and sensitivities

171 172

Table D1: Estimation of elasticities and sensitivities for each model parameter for Oenothera

173

glazioviana

174

Sensitivity for

Elasticity for

λ

σ d2

λ

σ d2

Flowering probability - p f (x)

-0.44

0.83

-0.11

0.23

Intercept

-18.27

0.0057

0.013

0.099

0.1

Slope rosette size x

6.91

0.014

0.031

0.093

0.094

Fecundity (number of seeds) - f (x)

0.00015

0.0008

0.25

1.00

Intercept

1.04

0.27

2.25

0.26

1.04

Slope rosette size x

2.22

0.80

7.22

1.69

7.12

Seed establishment probability - pe

21.6

241

0.25

1.00

Mean value

0.26

2.17

1.14

4.43

0.26

1.14

0.019

0.04

Annual survival of rosettes - s a (x)

1.51

-1.24

0.75

-0.53

Intercept

0.36

1.23

-0.082

0.42

-0.013

Slope rosette size x

0.17

0.87

-2.05

0.14

-0.15

Trait

Estimate

-4.61

Rosette size at one year old - f b ( y, x) Mean value

0.078

Rosette size at time t + 1 - f s ( y, x) Intercept

0.96

0.79

1.42

0.72

0.61

Slope rosette size x

0.59

1.05

1.38

0.59

0.36

Residual s.d.

0.67

0.41

3.00

0.17

0.60

175 176

12

Population viability analysis of plant and animal populations with stochastic integral projection models

177

Table D2: Estimation of elasticities and sensitivities for each model parameter for Zootoca vivipara Sensitivity for

Elasticity for

λ

σ d2

λ

σ d2

Breeding probability - pb ( x, Z )

0.30

0.13

0.26

0.22

Intercept

-29.9

0.0035

4.3e-4

0.11

0.028

Slope SVL x

0.59

0.15

0.019

0.10

0.025

Total clutch size - f (x)

0.050

0.031

0.26

0.35

λ

- Mean value

-1.41

0.036

0.027

0.057

0.084

θ

- Intercept

-2.09

0.18

0.13

0.43

0.6

θ

- Slope SVL x

0.063

11.7

8.4

0.82

1.18

Clutch success - cs(Z )

0.25

0.012

0.26

0.024

Mean value

0.016

6.7e-4

0.046

0.0039

0.49

0.32

0.26

0.34

0.12

0.08

0.012

0.016

Juvenile survival - s j

0.86

0.56

0.26

0.34

Mean value

0.17

0.11

0.19

0.24

Trait

Estimate

2.64

Female ratio - p f Mean value

-0.088

-0.98

SVL at one year old - f b ( y, x, Z ) Intercept

38.5

0.0016

6.3e-4

0.067

0.054

Residual variance

4.0e-4

0.066

0.19

2.9e-5

1.7e-4

1.1e-4

2.9e-4

2.9e-4

0.0016

Sub-adult and adult survival - s a

1.20

-0.11

0.74

-0.13

Mean value

0.30

-0.026

0.070

-0.012

Power exponent

2.494

0.21

SVL of older lizards - f s ( y, x, Z ) Intercept

47.82

0.016

0.011

0.85

1.12

Slope SVL x

0.25

0.85

0.61

0.24

0.34

Residual s.d.

7.94

8.0e-5

7.0e-4

7.0e-4

0.012

178 179

13

Population viability analysis of plant and animal populations with stochastic integral projection models

180

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181

Baron J-P, Le Galliard J-F, Tully T, Ferrière R (2010a) Cohort variation in offspring growth and

182

survival: prenatal and postnatal factors in a late-maturing viviparous snake. Journal of Animal

183

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