Physiologically structured population models: Formulation, analysis and ecological insights

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Physiologically structured population models:

Formulation, analysis and ecological insights

André M. De Roos

Institute for Biodiversity and Ecosystem Dynamics University of Amsterdam

in collaboration with

Lennart Persson

Ecology and Environmental Sciences

Umeå University, Sweden January 26, 2017

(2)

A central issue in ecology and evolution

Life history

0"

200"

400"

600"

800"

1000"

1200"

1990" 1995" 2000" 2005" 2010" 2015"

Monarch(bu+erﬂies((x(106)(

Year(

Monarch(bu+erﬂies(overwintering(in(Mexico(

Population growth

& demography

Life history evolution

0 20 40 60 80 100 120

1976 1980 1984 1988 1992 1996 2000 2004 2008 2012

Index (for unsmoothed data 1976=100)

United Kingdom United Kingdom

0 20 40 60 80 100 120

1976 1980 1984 1988 1992 1996 2000 2004 2008 2012

Index for unsmoothed data (1976=100)

United Kingdom

Population & Community dynamics

(3)

The conceptual issue:

Development like other complicating factors?

Unlike modulating factors such as:

§ Spatial heterogeneity

• Influence excluded through homogeneous mixing (chemostats)

§ Intraspecific genetic variation

• Influence excluded by using clonal or inbred individuals (parthenogenetic species, iso-female lines)

NO! Like reproduction and mortality, development is a constituent component of population dynamics

Development is next to mortality the most certain population dynamics process, reproduction is only secondary

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Individual characteristics &

traits Body size

Demographic models

Matrix or integral projection models

Individual life history Reproduction, Development,

Mortality

Approaches to study impacts of life history

Population dynamics Abundance, Size structure

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§ Matrix models:

discrete time, discrete individual states (i-states)

§ Integral projection models:

discrete time, continuous i-states

§ Characteristics:

• Data-driven, tight link with life history observations

• Non-mechanistic, no functional individual life history description x(l, t + 1) =

Z

⌦

0

B@ D(l,`)

| {z }

inheritance

fecundity

z}|{R(`) + G(l,`)

| {z }

growth

survival

z}|{S(`) 1

CAx(`, t)d`

Commonly used approaches for analysis

0 B@

x₁(t + 1) ...

x_n(t + 1) 1 CA =

0 B@

a₁₁ . . . a_1n ... . .. ... a_n1 . . . a_nn

1 CA

0 B@

x₁(t) ... x_n(t)

1 CA

H. Caswell, 2001. Matrix Population Models. Sinauer Associates

Ellner, Child & Rees, 2016. Data-driven Modelling of Structured Populations:

A Practical Guide to the Integral Projection Model. Springer

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A data-driven approach: IPMs

implementing the ipm

The next step is to write down the kernel and check whether its formulation matches our knowledge of the life cycle and the data collection protocols,

Kðz⁰;zÞ ¼sðzÞGðz⁰;zÞ þsðzÞpbðzÞprC0ðz⁰;zÞ=2: eqn 12 In this instance, both the survival and reproduction kernel components contain the s(z) term, because the main period of mortality occurs prior to reproduction.

The survival component includes the growth kernelG(z′, z); as in the example below,G is often specified in terms of the conditional mean, variance and distribution fam- ily of subsequent size. The reproduction component is simply the product of the reproduction function, pb(z), the probability of survival from spring until summer, p_r, and the conditional offspring size function, C₀(z′,z).

The factor of 1/2 appears because we are tracking the dynamics of females and have assumed an equal sex ratio.

We are going to use the approach given in Box 1, so we need to specify theP(z′,z) andF(z′,z) functions:

## Define the survival-growth kernel P z1z<- function (z1, z, m.par){

return(s z(z, m.par) * G_z1z(z1, z, m.par)) }

## Define the reproduction kernel

F z1z<- function (z1, z, m.par){

return( s z(z, m.par) * pb_z(z, m.par) * (1/2) * pr_z(m.par) * C_0z1(z1, z, m.par) )

}

These are just R translations of the two kernel components, and each function is passed a numeric vector,

m.par, that holds the parameter values for the underlying demographic regressions. In order to complete the imple- mentation of our model, we next need to define the various functions called within P_z1z and F_z1z. These follow in a very intuitive way from the statistical models we fitted to data. For example, the function describing the probability of survival is:

s z<- function(z, m.par){

linear.p<- m.par["surv.int"]+m.par["surv.z"] * z

# linear predictor

p<- 1/(1+exp(-linear.p))

# inv-logistic trans return(p)

}

pb z<- function(z, m.par){

linear.p<- m.par["repr.int"]+m.par["repr.z"] * z

p<- 1/(1+exp(-linear.p))

return(p) }

These functions extract the appropriate parameters, calculate the linear predictor at each value ofz and then transform this onto the probability scale. For s_z and Mass t, z

Probability of surviving

Mass t, z

Mass t+1, z’

Mass t, z

Probability of reproducing

2·0 2·5 3·0 3·5 2·0 2·5 3·0 3·5

2·0 2·5 3·0 3·5

0·00·20·40·60·81·00·00·20·40·60·81·0 2·02·53·03·52·02·53·03·5

Maternal mass t, z

Offspring mass t+1, z’

(a) (b)

(c) (d)

Fig. 2. The main mass-dependent demographic processes in the Soay sheep life cycle, as a function of current size,zin year t. (a) The probability of survival, (b) female mass in the next summer census (August catch), (c) the probability of reproduction and (d) offspring mass. Source file:

Ungulate Calculations.R in Sup- porting Information.

From life cycle to IPM kernel 535

Life history

Body mass

Density time t+1

Body mass

Density time t

Non-mechanistic representation of life history based on data

representative of current conditions

x(l, t+ 1) = Z

⌦

0 B@

reproduction

z }| {

D(l,`)R(`) + G(l,`)

| {z }

growth

survival

z}|{S(`) 1

CAx(`, t)d`

S(`)

R(`)

G(l,`)

D(l,`)

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Individual characteristics &

traits Body size

Demographic models:

Matrix or integral projection models

• Population rather than community dynamics (population growth rate)

• Energetics ignored (maintenance costs, mass/energy conservation)

Mortality

• Fixed development rates (age, stage), independent of food

Structured population models

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Life history’s most prominent feature:

Growth in body size (a doubling at least )

1 mm

x 10⁹

Intra-specific variation in body size!

All 3 perch are 4 years old!

Foto by Emma van der Woude

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• Community dynamics (resources, consumers, predators)

• Food-dependent growth in body size

• Strict conservation of mass/energy including maintenance costs

Size-structured population models based on individual energy budgets

Community dynamics Population

dynamics Population

dynamics Individual

characteristics &

traits Body size

Mortality

Structured population models

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Nonlinearity: population feedback on life history

Life history

0"

200"

400"

600"

800"

1000"

1200"

1990" 1995" 2000" 2005" 2010" 2015"

Monarch(bu+erﬂies((x(106)(

Year(

Monarch(bu+erﬂies(overwintering(in(Mexico(

Population growth

& demography

Life history evolution

0 20 40 60 80 100 120

1976 1980 1984 1988 1992 1996 2000 2004 2008 2012

Index (for unsmoothed data 1976=100)

United Kingdom United Kingdom

0 20 40 60 80 100 120

1976 1980 1984 1988 1992 1996 2000 2004 2008 2012

Index for unsmoothed data (1976=100)

United Kingdom

Population & Community dynamics

(11)

Physiologically Structured Population Models

Environmental state

• Resource density

• Predation risk

• Population density

Individual state

• age

• size/mass

• sex or genotype

x = ( x

₁

, … , x

_K

)

E = ( E

₁

, … , E

_N

)

Development Reproduction

Mortality Impact

g (x, E)

β ^(x, ^E)

µ ⁽ ^x, ^E)

γ ^(x, ^E)

Population abundance compositionand

Population impact

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Types of population feedback on life history

§ Number of conspecifics (direct density-dependence)

• Interference competition, e.g. for mates or nesting sites

• Phenomenological, non-mechanistic

§ Resource density (indirect)

• Exploitative foraging on shared resource

§ Predation risk (indirect)

• Top-down control on prey population through shared predator

Environment

§ Dynamic traits (i-state variables)

• Age, body size, energy reserves

§ Static traits (i-state variables)

• Sex, frailty, genotype

Individual

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Physiologically Structured Population Models

Environmental state

• Resource density

• Predation risk

• Population density

Individual state

• age

• size/mass

• sex or genotype

x = ( x

₁

, … , x

_K

)

E = ( E

₁

, … , E

_N

)

Development Reproduction

Mortality Impact

g (x, E)

β ^(x, ^E)

µ ⁽ ^x, ^E)

γ ^(x, ^E)

Population abundance compositionand

Population impact

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Model ingredients and parameterization

Body size (g)

0.001 0.01 0.1 1 10 100

Attack rate (L/s)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Body size (g)

0.001 0.01 0.1 1 10 100

0.0 0.5 1.0 1.5 2.0 2.5 3.0

_____ Roach 3.5

_____ Perch

0.5 mm cladoceran 1 mm cladoceran

On the basis of independent, individual-level

data and experiments

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Dynamic energy budget models:

Energy/mass conservation principle

Energy intake (assimilation from food)

Energy expenditure (maintenance, growth, reproduction) =

Energy intake

Somatic maintenance

Growth

Reproduction

§ “Kappa-rule” models

(Kooijman, 1993, 2002, 2009)

Reproduction proportional to ingestion

§ Net-production models

Maintenance covered first

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Dynamic energy budget model

Food

assimilation Metabolism Growth

Reproduction

κ 1 − κ

“_Kappa-rule” model or net-assimilation model

Net-production model

Food

assimilation Metabolism

Reproduction Growth

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Kooijman’s Dynamic Energy Budget theory

Reserve&E&

Structure&V& Maturity&

(juveniles)&

Reproduc7on&

(adults)&

Soma'c)

maintenance))!p_M

Food&X& Faeces&

Eggs&

Maturity)

maintenance))!p_J κ 1−κ

! p_C

Assimila'on)p!_A

Growth)

κ p!_C − p!_M Reproduc'on)

(1−κ⁾p!_C − p!_M

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Estimating DEB parameters from individual data

Feeding level 1 Feeding level 2 Feeding level 3

nieuwe analyse 6/7/12

Feeding level 1 Feeding level 2 Feeding level 3

20 25 30

shell length (mm)

600 800 1000 1200

cumulative eggs per snail

0 50 100 150

10 15 20

time (days)

shell length (mm)

0 50 100 150

0 200 400 600

time (days)

cumulative eggs per snail

yBA = 0.95

yBA = 0.55

yBA J_Am^a 

yBA yBA

adapted 6/7/12 adapted 6/7/12

fe = 0.5, 130 uL/mg, WV0=0.25 ug dwt, for core model 0.01 ug

1

mm)

initial pars.

adapted pars.

3 3.5

0.6 0.8

metric length (

data 1 data 2 birth 2

2.5

se (µL O2/d)

0.2 0.4

embryo volum

0.5 1 1.5

oxygen u

0 2 4 6 8 10

0

time (days)

e

0 2 4 6 8 10

0 0.5

time (days)

adapted 6/7/12 adapted 6/7/12

fe = 0.5, 130 uL/mg, WV0=0.25 ug dwt, for core model 0.01 ug

1

mm)

initial pars.

adapted pars.

3 3.5

0.6 0.8

metric length ( data 1

data 2 birth 2

2.5

se (µL O2/d)

0.2 0.4

embryo volum

0.5 1 1.5

oxygen u

0 2 4 6 8 10

0

time (days)

e

0 2 4 6 8 10

0 0.5

time (days)

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@c(t, s)

@t + @g(s, R)c(t, s)

@s = µ(s, R)c(t, s) g(s_b, R)c(t, s_b) =

Z s_m sb

(s, R)c(t, s)ds dR

dt = G(R)

Z sm

s_b

(s, R)c(t, s)ds

A generic size-structured model

§ Individuals are born with size (mass)

§ Growth rate in body size:

§ Reproduction rate:

§ Resource intake rate:

§ Mortality rate:

• “Sinko & Streifer model”

(Ecology, 1967)

• VonFoerster (1959)

• Frederickson et al. (1967)

• Bell & Anderson (1967)

• Metz & Diekmann (1986) Springer Lecture Notes in Biomathematics 68

s

_b

g(s, R)

(s, R)

µ(s, R)

(20)

b

t a

time

size Population birth rate

t

Newborns

Survivors Contribution to birth rate

x(a)

F(a)b(t a) (x(a))F(a)b(t a)

Integro-delay-differential equation formulation

b

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The general structured model

b(t) =

Z ₁

0

Fecundity

z }| {

(⌅(a, E_t), E(t)) F(a, E_t)

| {z } Survival

b(t a)da

I(t) =

Z ₁

0

Impact

z }| {

(⌅(a, E_t), E(t)) F(a, E_t)

| {z } Survival

b(t a)da

E(t) =

X(t) Y (t)

!

X(t) = G_X (X(t), Y (t), I(t)), dY

dt = G_Y (X(t), Y (t), I(t))

Needed: ⌅(a, E_t), the solution of d⇠

d⌧ = g(⇠(⌧), E(t a+⌧)), 0 < ⌧ < a, ⇠(0) 2 ⌦_b

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Functional life history representations

Reserve&E&

Structure&V& Maturity&

(juveniles)& Reproduc7on&

(adults)&

Soma'c)

maintenance))!p_M

Food&X& Faeces&

Eggs&

Maturity) maintenance))!p_J κ 1−κ

! p_C

Assimila'on)p!_A

Growth)

κp!_C − p!_M Reproduc'on)

(1−κ⁾p!_C − p!_M

Kooijman’s Dynamic Energy Budget model

How can we compute model equilibria

given a mildly complex life history model?

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Steady state analysis

http://www.elsevier.com/locate/ytpbi Theoretical Population Biology 63 (2003) 309–338

Steady-state analysis of structured population models

O. Diekmann,^a M. Gyllenberg,^b,^! and J.A.J. Metz^c,d

aDepartment of Mathematics, University of Utrecht, P.O. Box 80010, 3580 TA Utrecht, The Netherlands

bDepartment of Mathematics, University of Turku, 20014 Turku, Finland

cInstitute for Evolutionary and Ecological Sciences, Leiden University, Kaiserstraat 63, NL-2311 GP Leiden, The Netherlands

dAdaptive Dynamics Network, IIASA, A-2361 Laxenburg, Austria Received 12 March 2002

Abstract

Our systematic formulation of nonlinear population models is based on the notion of the environmental condition. The defining property of the environmental condition is that individuals are independent of one another (and hence equations are linear) when this condition is prescribed (in principle as an arbitrary function of time, but when focussing on steady states we shall restrict to constant functions). The steady-state problem has two components: (i) the environmental condition should be such that the existing populations do neither grow nor decline; (ii) a feedback consistency condition relating the environmental condition to the community/population size and composition should hold. In this paper we develop, justify and analyse basic formalism under the assumption that individuals can be born in only finitely many possible states and that the environmental condition is fully characterized by finitely many numbers. The theory is illustrated by many examples. In addition to various simple toy models introduced for explanation purposes, these include a detailed elaboration of a cannibalism model and a general treatment of how genetic and physiological structure should be combined in a single model.

Keywords: Population dynamics; Physiological structure; Nonlinear; Feedback via the environment; Deterministic at population level; Cannibalism;

Finitely many states at birth; Population genetics; Adaptive dynamics; Competitive exclusion

1. Introduction 1.1. Aim of the paper

The aim of this paper is to provide a technical framework for analysing the existence (as well as dependence on parameters) of steady states for very large classes of population models, including models for interacting physiologically structured populations. In particular, we show that the steady-state problem can be brought into the form

b¼LðIÞb;

I ¼GðIÞb;

(

ð1:1Þ

where

(i) bis a vector ofbirth rates(each componentb_jofb is the steady rate at which individuals are born with the state at birth numbered j).

(ii) Iis a vector describing theenvironmental conditions as far as they are influenced by interaction. The defining property is that individuals are independent of one another whenIis given (in general as a function of time; in this paper we restrict to constantI).

(iii) L(I) is the next-generation matrix. The (i, j)- element ofL(I) is the expected number of offspring with birth statei born to an individual that itself was born with statej, given steady environmental conditions as specified byI.

(iv) G(I) is the feedback matrix. The (i, j)-element of G(I) gives the lifetime contribution to the ith componentIiofIof an individual born with state j, given steady environmental conditions as specified byI.

Following up on earlier work (Diekmann et al., 2001;

Kirkilionis et al., 2001) we include an operational description of how to derive L(I) and G(I) from more basic modelling ingredients, such as maturation-, death-, and birth-rates. Moreover, we develop

!Corresponding author. Fax:+358-40-333-65-95.

E-mail address:[email protected] (M. Gyllenberg).

doi:10.1016/S0040-5809(02)00058-8

July 13, 2001 15:28 WSPC/103-M3AS 00126

Mathematical Models and Methods in Applied Sciences Vol. 11, No. 6 (2001) 1101–1127

⃝c World Scientific Publishing Company

NUMERICAL CONTINUATION OF EQUILIBRIA OF PHYSIOLOGICALLY STRUCTURED

POPULATION MODELS. I. THEORY

MARKUS A. KIRKILIONIS^∗

Interdisziplinäres Institut für Wissenschaftliches Rechnen, Universität Heidelberg, Im Neuenheimer Feld 368,

69120 Heidelberg, Germany

ODO DIEKMANN

Department of Mathematics, University of Utrecht,

Budapestlaan 6, P. O. Box 80010, 3508 TA Utrecht, The Netherlands

BERT LISSER, MARGREET NOOL and BEN SOMMEIJER Centrum voor Wiskunde en Informatica,

P. O. Box 94079, 1090 GB Amsterdam, The Netherlands

ANDRE M. DE ROOS

Faculty of Biology, University of Amsterdam, Kruislaan 320, 1098 SM Amsterdam, The Netherlands

Communicated by N. Bellomo Received 2 May 2000 Revised 12 November 2000

The paper introduces a new numerical method for continuation of equilibria of models describing physiologically structured populations. To describe such populations, we use integral equations coupled with each other via interaction (or feedback) variables.

Additionally we allow interaction with unstructured populations, described by ordinary diﬀerential equations. The interaction variables are chosen such that if they are given functions of time, each of the resulting decoupled equations becomes linear. Our numerical procedure to approximate an equilibrium which will use this special form of the underlying equations extensively. We also establish a method for local stability analysis of equilibria in dependence on parameters.

Keywords: Physiologically structured population models, numerical continuation of equilibria

∗E-mail: [email protected]; corresponding author.

1101

July 13, 2001 15:28 WSPC/103-M3AS 00126

69120 Heidelberg, Germany ODO DIEKMANN

Budapestlaan 6, P. O. Box 80010, 3508 TA Utrecht, The Netherlands BERT LISSER, MARGREET NOOL and BEN SOMMEIJER

Centrum voor Wiskunde en Informatica, P. O. Box 94079, 1090 GB Amsterdam, The Netherlands

ANDRE M. DE ROOS

1101

July 13, 2001 15:28 WSPC/103-M3AS 00126

69120 Heidelberg, Germany ODO DIEKMANN

Budapestlaan 6, P. O. Box 80010, 3508 TA Utrecht, The Netherlands BERT LISSER, MARGREET NOOL and BEN SOMMEIJER

Centrum voor Wiskunde en Informatica, P. O. Box 94079, 1090 GB Amsterdam, The Netherlands

ANDRE M. DE ROOS

1101

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Equilibrium analysis

§ Individuals are assumed to live in an environment characterized by a (finite) set of environment variables:

Environment variables can include independent quantities like resource density and density of predators, but also density-

dependent measures like total number of individuals or biomass in the population

§ Individuals are characterized by their individual or i-state, which is a (finite) set of physiological characteristics (traits such as age, size, sex, energy reserves):

§ Individual and environmental state variables determine the

individual life history (development, impact, reproduction, mortality)

x = (x

₁

, ... , x

_k

) 2 ⌦ ⇢ R

^k

E = (E

₁

, ... , E

_m

) 2 R

^m

(25)

§ Development follows a deterministic process that is continuous in time:

§ Individuals have an impact on their environment

§ Reproduction is a function of individual and environment state

§ Mortality is a function of individual state and environment state

§ Individuals are born with an i-state that is one of a finite set of possible states at birth:

with each potential state at birth a valid i-state:

b

b 2 { 1, ... , _m}

The generic model for individual life history

j = ( _j₁, ... , _jk) 2 ⌦ ⇢ R^k

j

dx

da = g(x,E) (x, E) (x,E)

µ(x,E)

(26)

Environment dynamics

§ Environment variables may in isolation follow autonomous dynamics:

or may be functions of the population (to model direct density dependence):

dE

dt = G(E)

E(t) = Z

⌦

(x,E)n(t,x)dx

(27)

§ In equilibrium of the structured population the expected lifetime reproduction equals 1:

§ In addition, the dynamics of the environment should be balanced by the impact of the population:

or

Computing an equilibrium of a structured model

How to compute these integrals?

˜b

Z ₁

0

(x(a,E˜),E˜)F(a,E˜)da

| {z }

Lifetime impact on environment

= G⇣ E˜⌘ Z ₁

0

(x(a,E˜),E˜)F(a, E)˜ da

| {z }

Expected lifetime reproduction

= 1

˜b

Z ₁

0

(x(a,E˜),E˜)F(a,E˜)da

| {z }

Lifetime impact on environment

= ˜E

(28)

Computing an equilibrium of a structured model

§ Probability of survival up to age a:

§ Cumulative number of offspring up to age a:

§ Cumulative impact up to age a:

F (a, E ˜ ) = exp

✓ Z

_a

0

µ ⇣

x(↵, E) ˜ ⌘ d↵

◆ H (a, E ˜ ) =

Z

a 0

(x(↵, E ˜ ), E ˜ ) F (↵, E) ˜ d↵

I (a, E ˜ ) =

Z

a 0

(x(↵, E ˜ ), E ˜ ) F (↵, E ˜ ) d↵

(29)

Equilibrium conditions

H ( 1 , E ˜ ) =

Z

₁

0

(x(↵, E ˜ ), E) ˜ F (↵, E ˜ ) d↵ = 1

˜ b I ( 1 , E ˜ ) = ˜ b

Z

₁

0

(x(↵, E ˜ ), E) ˜ F (↵, E ˜ ) d↵ = G( ˜ E)

(30)

§ Differentiating the probability of survival up to age leads to:

§ Differentiating with respect to a yields:

The computational approach

F(a, ˜E)

dF

da = µ(x(a,E˜),E˜)F(a,E˜), F(0,E˜) = 1

H(a,E˜)

dH

da = (x(a,E˜), E)˜ F(a,E˜), H(0, E) = 0˜

dI

da = (x(a,E˜),E˜)F(a, E),˜ I(0,E˜) = 0 I(a,E)˜

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§ The equilibrium of a non-linear structured population model is determined by:

which has to be solved (numerically and iteratively) for the unknowns and .

§ The values of and are evaluated by integration of the ODEs:

E˜ b˜

H(1, ˜E) = 1

b I˜ (1, ˜E) = G( ˜E)

Putting it all together

H(1, ˜E) I(1, ˜E) 8>

>>

<

>>

>: dx

da = g(x(a, ˜E), ˜E), x(0, ˜E) = _b dF

da = µ(x(a, ˜E), ˜E)F(a, ˜E), F(0, ˜E) = 1 dH

da = (x(a, ˜E), ˜E)F(a, ˜E), H(0, ˜E) = 0 dI

da = (x(a, ˜E), ˜E)F(a, ˜E), I(0, ˜E) = 0

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Environmental,state

• Resource(density

• Predation(risk

• Population(density

Individual,state

• age

• size/mass

• sex(or(genotype

) , ,

( x

₁

! x

_K

x =

) ,

,

( E

₁

! E

_N

E =

Mortality Reproduction

Impact

dx/dt = g (x, E)

) , ( x E µ

β ⁽ ^x, ^E)

) , ( x E γ

Development

PSPManalysis _A ^Ma

tla b/R /C p ac ka ge

Demographic analysis

• Population growth rate

• Sensitivity to model parameters

• Stable population distribution

• Reproductive value

Equilibrium analysis

• Equilibrium states dependent on parameters

• Detection of bifurcation points (saddle-node, transcritical)

• Continuation of bifurcation points dependent on 2 parameters

Evolutionary analysis

• Adaptive dynamics approach

• Detection and classification of Evolutionary Singular Strategies (ESS)

• ESS continuation dependent on parameters

• Computation of Pairwise Invasibility Plot (PIP)

• Evolutionary dynamics (canonical equation)

Physiologically Structured Population Models: Analysis

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A size-structured, tritrophic example

Size (s)

Density

Consumer

s_b s_m

Resource

µ^(s,^P)

γ^(s,^R)

g(s,R) β^(s,^R)

Predator

• Ingestion scales allometrically with size

• Adults continue growing, while reproducing

• Food-dependent growth and reproduction

• Maturation when reaching size threshold

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Predator per capita growth rate:

Consumer growth rate in size:

Consumer fecundity:

Consumer mortality:

Consumer foraging:

Resource turnover:

A size-structured, tritrophic example

Size (s)

Density

Consumer

s_b s_m

Resource

µ^(s,^P)

γ^(s,^R)

g(s,R) β^(s,^R)

Predator

µ(s, P) = 8<

:

µ_b+ aP

1 + aT_hB if s < s_V

µ_b otherwise

B 1 + aT_hB

⇢(R_max R) g(s, R) = ⌫

✓

s_m R

R_h + R s

◆

(s, R) = r_m R

R_h + Rs² if s s_j

(s, R) = I_m R

R_h + Rs²

(35)

Consumer fecundity:

Consumer mortality:

Consumer foraging:

Resource turnover:

A size-structured, tritrophic example

Size (s)

Density

Consumer

s_b s_m

Resource

µ^(s,^P)

γ^(s,^R)

g(s,R) β^(s,^R)

Predator

µ(s, P) = 8<

:

µ_b+ aP

µ_b otherwise

B 1 + aT_hB

⇢(R_max R) g(s, R) = ⌫

✓

s_m R

R_h + R s

◆

(s, R) = r_m R

R_h + Rs² if s s_j

(s, R) = I_m R

R_h + Rs²

void Development(int lifestage[POPULATION_NR], double *istate[POPULATION_NR],

double *birthstate[POPULATION_NR], int BirthStateNr, double E[],

double development[POPULATION_NR][I_STATE_DIM]) {

// Implement the development function below development[0][0] = NU*(SM*R/(R + RH) - S);

return;

}

(36)

Consumer fecundity:

Consumer mortality:

Consumer foraging:

Resource turnover:

A size-structured, tritrophic example

Size (s)

Density

Consumer

s_b s_m

Resource

µ^(s,^P)

γ^(s,^R)

g(s,R) β^(s,^R)

Predator

µ(s, P) = 8<

:

µ_b+ aP

µ_b otherwise

B 1 + aT_hB

⇢(R_max R) g(s, R) = ⌫

✓

s_m R

R_h + R s

◆

(s, R) = r_m R

R_h + Rs² if s s_j

(s, R) = I_m R

R_h + Rs²

void Mortality(int lifestage[POPULATION_NR], double *istate[POPULATION_NR], double *birthstate[POPULATION_NR], int BirthStateNr, double E[],

double mortality[POPULATION_NR]) {

// Implement the mortality function below if (lifestage[0] == 0)

mortality[0] = MUB + A*P/(1+A*TH*B);

else

mortality[0] = MUB;

return;

}

(37)

Equilibrium analysis of non-linear, density dependent models

10⁻⁶ 10⁻⁵ 10⁻⁴

Predator

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³

Consumer

0.5 1 1.5 2 2.5 3 3.5 4

x 10⁻⁴ 10⁻⁵

10⁻⁴

Maximum resource density

Resource density

1. Starting from a trivial equilibrium

>> [data1, rep1, bp1, bt1] = ...

PSPMequi(’Test', 'EQ', [1.0E-06 1.0E-06],..);

(38)

Equilibrium analysis of non-linear, density dependent models

10⁻⁶ 10⁻⁵ 10⁻⁴

Predator

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³

Consumer

0.5 1 1.5 2 2.5 3 3.5 4

x 10⁻⁴ 10⁻⁵

10⁻⁴

Resource density BP #0

>> [data1, rep1, bp1, bt1] = ...

2. Starting from the detected consumer invasion point

>> [data2, rep2, bp2, bt2] =

PSPMequi(’Test', 'EQ', bp1([1 2 5]),..);

(39)

Equilibrium analysis of non-linear, density dependent models

10⁻⁶ 10⁻⁵ 10⁻⁴

Predator

0.5 1 1.5 2 2.5 3 3.5 4

x 10⁻⁴ 10⁻⁵

10⁻⁴

BPE #1

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³

Consumer

BPE #1 BPE #1

>> [data1, rep1, bp1, bt1] = ...

>> [data2, rep2, bp2, bt2] =

3. Starting from the detected predator invasion point

>> [data3, rep3, bp3, bt3] =

PSPMequi(’Test', 'EQ', bp2([1 2 3 7 5]),..);

(40)

Equilibrium analysis of non-linear, density dependent models

0.5 1 1.5 2 2.5 3 3.5 4

x 10⁻⁴ 10⁻⁵

10⁻⁴

BPE #1 LP

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³

Consumer

BPE #1 BPE #1

LP LP

10⁻⁶ 10⁻⁵ 10⁻⁴

Predator

LP

>> [data1, rep1, bp1, bt1] = ...

>> [data2, rep2, bp2, bt2] =

3. Starting from the detected predator invasion point

>> [data3, rep3, bp3, bt3] =

PSPMequi(’Test', 'EQ', bp2([1 2 3 7 5]),..);

(41)

0.5 1 1.5 2 2.5 3 3.5 4 x 10⁻⁴ 10⁻⁵

10⁻⁴

BPE #1 LP

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³

Consumer

BPE #1 BPE #1

LP LP

10⁻⁶ 10⁻⁵ 10⁻⁴

Predator

LP

1 2 3 4

x 10⁻⁴ 0.01

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Consumer mortality

1 2 3 4

x 10⁻⁴ 0.01

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Consumer mortality

1 2 3 4

x 10⁻⁴ 0.01

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Consumer mortality

1 2 3 4

x 10⁻⁴ 0.01

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Consumer mortality

1. Continuing consumer invasion boundary

>> [data4, rep4] = ...

PSPMequi(’Test’,’BP’,[bp1([1 2]) 0.01],..);

2. Continuing predator invasion boundary

>> [data5, rep5] =

PSPMequi(’Test’,’BPE’,[bp2([1 2 5]) 0.01],..);

3. Continuing predator persistence boundary

>> [data6, rep6] =

PSPMequi(’Test’,’LP’,[bp3([1:5] 0.01]),..);

Equilibrium analysis of non-linear,

density dependent models

(42)

Evolutionary Stable Strategy (ESS)

§ Selection gradient:

(computed numerically, while computing ecological equilibrium as function of a life history parameter)

§ Evolutionary endpoint:

§ Evolutionary dynamics: (Canonical equation of adaptive dynamics):

d R

⁰

dq

_q=q

res

= d dq

Z

₁

0

(s(a, R), ˜ R) ˜ p(a, s( · , R), ˜ R) ˜ da

q=qres

d R

⁰

dq

_q=q

res

= 0

dq_res

d⌧ / µ(q_res) ⁰ (q_res)

2 ˜b(q_res) dR⁰

dq _q=q

res

(43)

Adaptive Dynamics

§ Evolution in environment set by ecological dynamics

• Resident-mutant interaction: Resident sets the environment, , which determines the mutant’s fitness

§ Separation of evolutionary and ecological timescale

• Mutation limitation: Convergence to (new) ecological equilibrium between mutation events

• Domination or demise: Positive fitness (mutant growth) results in take-over, negative fitness leads to mutant disappearance R(q˜ _res)

q_mut,+

q_mut,- q_mut,-q_res q_mut,-q_res q_mut,+q_res q_mut,+q_res

Life history trait (parameter)

R⁰

R⁰ = 1

Physiologically structured population models: Formulation, analysis and ecological insights

Physiologically structured population models:

Formulation, analysis and ecological insights

A central issue in ecology and evolution

The conceptual issue:

Development like other complicating factors?

Demographic models

Approaches to study impacts of life history

§ Matrix models:

§ Integral projection models:

§ Characteristics:

Commonly used approaches for analysis

A data-driven approach: IPMs

Demographic models:

Matrix or integral projection models

Structured population models

Life history’s most prominent feature:

Growth in body size (a doubling at least )

Intra-specific variation in body size!

All 3 perch are 4 years old!

Size-structured population models based on individual energy budgets

Structured population models

Nonlinearity: population feedback on life history

Physiologically Structured Population Models

x = ( x

, … , x

)

E = ( E

, … , E

)

g (x, E)

β (x, E)

µ ( x, E)

γ (x, E)

Types of population feedback on life history

Environment

Individual

Physiologically Structured Population Models

x = ( x

, … , x

)

E = ( E

, … , E

)

g (x, E)

β (x, E)

µ ( x, E)

γ (x, E)

Model ingredients and parameterization

On the basis of independent, individual-level

data and experiments

Dynamic energy budget models:

Energy/mass conservation principle

Energy intake (assimilation from food)

Energy expenditure (maintenance, growth, reproduction) =

Reproduction proportional to ingestion

Maintenance covered first

Dynamic energy budget model

κ 1 − κ

Kooijman’s Dynamic Energy Budget theory

Reserve&E&

Structure&V& Maturity&

Reproduc7on&

Food&X& Faeces&

Eggs&

Estimating DEB parameters from individual data

A generic size-structured model

§ Individuals are born with size (mass)

§ Growth rate in body size:

§ Reproduction rate:

§ Resource intake rate:

§ Mortality rate:

s

g(s, R)

(s, R)

(s, R)

µ(s, R)

Integro-delay-differential equation formulation

The general structured model

Functional life history representations

β ^(x, ^E)

µ ⁽ ^x, ^E)

γ ^(x, ^E)

β ^(x, ^E)

µ ⁽ ^x, ^E)

γ ^(x, ^E)

β ⁽ ^x, ^E)

PSPManalysis _A ^Ma