Physiologically structured population models: Formulation, analysis and ecological insights

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 24, 2017

A short biography: André de Roos

http://staff.fnwi.uva.nl/a.m.deroos

§ Ecologist with a strong interest understanding how ecological systems function (dynamics) and

(some) mathematical skills

§ PhD in Theoretical Biology at Leiden University (1989) Supervisors: Hans Metz and Odo Diekmann

Topic: Numerical methods of physiologically structured population models (Escalator Boxcar Train)

an EBT implementation with adaptive spacing of cohort introduction times .... offers the best overall performance for SSPMs Zhang, Dieckmann, Brännström, 2017

§ Nowadays: using state-of-the-art (numerical) toolbox (dynamics, bifurcation analysis, adaptive dynamics) for studying dynamics of structured population models (PSPMs) to answer ecological and evolutionary questions

§ In case of PSPMs biology has driven the mathematical progress Do not blindly apply existing methods from mathematics or physics, think carefully about your biological system first

Overview of lectures

Population and Community Ecology of Ontogenetic Development

André M. de Roos & Lennart Persson

Princeton Monographs 51 André de Roos Lennart Persson

• Lecture 1: Conceptual, (somewhat) mathematical, the basic idea

• Lecture 2: Counterintuitive implications for ecological community structure

• Lecture 3: General formulation of a PSPM, numerical tools and techniques

• Lecture 4: Implications for ecological dynamics

Ecology is all about interactions

Modeling ecological dynamics

Modeling ecological dynamics

X₁

X₂ X₃

X₄

X₅ X₆

X₇ X₈

X₉ X₁₀ X₁₁

X₁₂

X₁₃ X₁₄

Alfred Lotka Vito Volterra

dx

dt =g (x

, . . . , x

)

i = 1, . . . , n

Dynamics of interacting populations

“Population dynamics: the variations in time and space in the sizes and densities of populations (the numbers of individuals per unit area)”

M.Begon, C.R.Townsend, J.L.Harper (2005) Ecology: From Individuals to Populations, Wiley-Blackwell

Is there a problem?

Populations considered as collections of elementary particles, only increasing and decreasing in abundance through

reproduction and mortality, respectively

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: dR

dt = p(R) f(R)C dC

dt = ✏f(R)C µC

A “general” predator-prey model

Two fundamental predictions:

§ Equilibrium consumer density decreases with increasing consumer mortality

§ Limit cycles occur under a wide range of conditions if (e.g. exponential or logistic prey growth)

Considered to be “general” because it “captures the bare essence of the predator-prey interaction”

p⁰(R) 0 f⁰(R) > 0

Food chains: trophic cascades

Longer food chains due to:

§ Increased resource productivity

§ Decreased exploitation of top predator

Predator-prey dynamics: cycles abound....

30% of 700 populations cycle (mainly fish and mammals)

... or perhaps not?

Kendall et al. Ecol. Letters1: 160-164 (1998)

Classic predator-prey cycles are not common

Murdoch et al. (2002) Nature417: 541-543

Distribution of cycles among classes defined by scaled period.

• SGC, single-generation cycles (

!

• DFC, delayed-feedback cycles (2 ≤ ! ≤ 4)

• CRC, consumer–resources cycles (period in years 4 T_C + 2 T_R )

No cycles fall in the intermediate class (INT) between single-species and consumer–resource cycles

The majority of observed population

cycles are not predator-prey cycle

A “general” predator-prey model

These unstructured, Lotka-Volterra-type models underlie our thinking and theory about ecological systems

But these models completely ignores the bare essence of life:

§ Individuals have to develop and growth in size before they can contribute to further population growth

(juvenile-adult stage structure, juvenile delay)

§ Mere existing costs energy (maintenance costs)

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: dR

dt = p(R) f(R)C dC

dt = ✏f(R)C µC f⁰(R) > 0

Today’s special

How juvenile-adult stage-structure and maintenance requirements overturn fundamental ecological

insights derived from unstructured models

p⁰(R)  0 and f⁰(R), g⁰(R) > 0 8>

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>>

: dR

dt = p(R) f(R)C dC

dt = g(R)C µC

The archetypical consumer-resource model

Two fundamental predictions:

§ Equilibrium consumer density decreases with increasing mortality

§ Equilibrium is always stable (no limit cycles)

Simple stage-structure

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: dR

dt = p(R) f_J(R)C_J f_A(R)C_A dC_J

dt = g_A(R)C_A g_J(R)C_J µ_JC_J dC_A

dt = g_J(R)C_J µ_AC_A

p⁰(R)  0 and f_J⁰ (R), f_A⁰ (R), g_J⁰ (R), g_A⁰ (R) > 0

When structure does not matter at all

f_J(R) = ↵_Jf(R), f_A(R) = ↵_Af(R) g_J(R) = g(R), g_A(R) = g(R) µ_J = µ_A = µ

dC

dt = g_A(R)C_A µ_JC_J µ_AC_A

= g_A(R)(1 z)C µ_JzC µ_A(1 z)C Define C = C_J + C_A and z = C_J/(C_J + C_A)

When structure does not matter at all

dC

dt = g_A(R)C_A µ_JC_J µ_AC_A

= g_A(R)(1 z)C µ_JzC µ_A(1 z)C Define C = C_J + C_A and z = C_J/(C_J + C_A)

dz

dt = 1

C_J + C_A

dC_J dt

C_J C_J + C_A

1 C_J + C_A

d(C_J + C_A) dt

= g_A(R)(1 z) g_J(R)z µ_Jz

z (g_A(R)(1 z) µ_Jz µ_A(1 z))

= g_A(R)(1 z)² g_J(R)z (µ_J µ_A)z(1 z)

When structure does not matter at all

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: dR

dt = p(R) (↵_Jz + ↵_A(1 z))f(R)C dC

dt = g(R)(1 z)C µC dz

dt = (1 z)² z g(R)

(1 z)¯ ² z¯ = 0 ) z¯ = 0

@1 + 2

s✓

1 + 2

◆2

1 1 A For t ! 1 z will always approach its equilibrium value ¯z:

When structure does not matter at all

Necessary assumptions:

§ Ingestion and production (maturation, reproduction) have the same dependence on resource density (but not stage-independent)

§ Mortality rate is the same for both stages 8>

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: dR

dt = p(R) ↵f¯ (R)C dC

dt = ¯g(R)C µC

in which ¯↵ = ↵_Jz¯ + ↵_A (1 z) and ¯ =¯ (1 z).¯

Generalizable to arbitrarily many stages!

p⁰(R)  0 and f_J⁰ (R), f_A⁰ (R) > 0 (but not f_J(R) / f_A(R))

The effects of stage-structure alone

§ Common assumption: numerical response is proportional to functional response:

8>

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: dR

dt = p(R) f_J(R)C_J f_A(R)C_A dC_J

dt = f_A(R)C_A f_J(R)C_J µ_JC_J dC_A

dt = f_J(R)C_J µ_AC_A

Unique equilibrium state

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:

H₁( ¯R,C¯¯_J, C¯_A) = p( ¯R) f_J( ¯R) ¯C_J f_A( ¯R) ¯C_A = 0 H₂( ¯R,C¯_J, C¯_A) = f_A( ¯R) ¯C_A f_J( ¯R) ¯C_J µ_JC¯_J = 0 H₃( ¯R,C¯_J, C¯_A) = f_J( ¯R) ¯C_J µ_AC¯_A = 0

)

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<

>>

>>

>>

>:

f_J( ¯R) f_A( ¯R) µ_A µ_Jµ_A = 0 C¯_J = p( ¯R)

f_J( ¯R) 1 + f_A( ¯R)/µ_A C¯_A = p( ¯R)

µ_A + f_A( ¯R)

The effects of stage-structure alone

J = 0 BB B@

p⁰( ¯R) f_J⁰ ( ¯R) ¯C_J f_A⁰ ( ¯R) ¯C_A f_J( ¯R) f_A( ¯R) f_A⁰ ( ¯R) ¯C_A f_J⁰ ( ¯R) ¯C_J f_J( ¯R) µ_J f_A( ¯R)

f_J⁰ ( ¯R) ¯C_J f_J( ¯R) µ_A

1 CC CA In a 3-dimensional ODE system a unique, positive equilibrium state ensures that D = detJ < 0

D = f_A( ¯R) + µ_A µ_Jf_J⁰ ( ¯R) ¯C_J + f_J( ¯R)f_A⁰ ( ¯R) ¯C_A)

The effects of stage-structure alone

§ Changes in equilibrium density can be assessed using the implicit function theorem:

0 BB BB BB B@

@H₁

@R

@H₁

@C¯_J

@H₁

@C¯_A

@H₂

@R

@H₂

@C¯_J

@H₂

@C¯_A

@H₃

@R

@H₃

@C¯_J

@H₃

@C¯_A 1 CC CC CC CA

0 BB BB BB BB

@

dR¯ dµ_J dC¯_J dµ_J dC¯_A

dµ_J 1 CC CC CC CC A

+ 0 BB BB

@ 0 C¯_J

0 1 CC CC

A = 0

) J 0 BB BB BB BB

@

dR¯ dµ_J dC¯_J dµ_J dC¯_A

dµ_J 1 CC CC CC CC A

= 0 BB BB

@ 0 C¯_J

0 1 CC CC A

The effects of stage-structure alone

dC¯_J

dµ_J = D ¹

p⁰( ¯R) f_J⁰ ( ¯R) ¯C_J f_A⁰ ( ¯R) ¯C_A 0 f_A( ¯R) f_A⁰ ( ¯R) ¯C_A f_J⁰ ( ¯R) ¯C_J C¯_J f_A( ¯R)

f_J⁰ ( ¯R) ¯C_J 0 µ_A

= D ¹ µ_A p⁰( ¯R) + f_J⁰ ( ¯R) ¯C_J + f_A⁰ ( ¯R) ¯C_A + f_J⁰ ( ¯R)f_A( ¯R) ¯C_J C¯_J

) dC¯_J

dµ_J < 0

dC¯_A

dµ_J , dC¯_J

dµ_A and dC¯_A

dµ_A are also negative.

Characteristic equation

Analysis is complex, but checking the Routh-Hurwitz criteria reveals that the equilibrium is always stable

p⁰( ¯R) f_J⁰ ( ¯R) ¯C_J f_A⁰ ( ¯R) ¯C_A f_J( ¯R) f_A( ¯R) f_A⁰ ( ¯R) ¯C_A f_J⁰ ( ¯R) ¯C_J f_J( ¯R) µ_J f_A( ¯R)

f_J⁰ ( ¯R) ¯C_J f_J( ¯R) µ_A

= 0

Stage structure alone does not invalidate the fundamental

predictions from unstructured consumer resource cycles,

if the numerical and functional response are proportional

Fecundity

(Net-production DEB models)

Maintenance

Maintenance takes precedence over production

Functional response

Resource

Numerical

response

Maintenance: implicit without stage structure

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: dR

dt = p(R) f(R)C dC

dt = ( f(R) T)⁺ C µ C

In the close neighbourhood of an equilibrium state, necessarily:

( f(R) T)⁺ = f(R) T

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: dR

dt = p(R) f(R)C dC

dt = f(R)C (µ + T)C

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: dR

dt = P f(R) (C_J + C_A) dC_J

dt = (✏f(R) T_A)⁺ C_A (✏f(R) T_J)⁺ C_J µ C_J dC_A

dt = (✏f(R) T_J)⁺ C_J µ C_A

Maintenance and stage structure together

§ In equilibrium either juvenile or adult consumer density can increase with increasing mortality (stage-independent,

juvenile, adult)

§ Population cycles can occur as a consequence of stage-

structured dynamics

P = ✏ = T_A = 1, T_J = 0.75 P = ✏ = T_A = 1, T_J = 1.5

Maintenance and stage structure together

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1.0

0 0.2 0.4 0.8 0.6 1.0

Mortality rate μ

Juv enile - A dult c onsumer densit y

Adults

Maintenance and stage structure together

0 1.0 1.5 2.0

0.5

0.5

0 0.1 0.2 0.3 0.4 0.6 0.7

Juvenile/adult maintenance ratio (T J /T A )

Mortality/maintenance ratio (μ/T_A )

Adult density

overcompensation

Juvenile density overcompensation

Maintenance and stage structure together

Assimilation increase ≈ 9% Maturation increase ≈ 18%

Reproduction increase ≈ 95%

Net increase in reproduction >> Net increase in maturation

Adult maintenance costs Juvenile maintenance costs

Resource density

Assimilation - Fecundity - Maturation

Fecundity Assimilation

Maturation

A₀+∆A A₀

M₀+∆M M₀

F₀+∆F F₀

Maintenance and stage structure together

0 20 40 60 80 100

Time

2.5

0 0.5 1.0 1.5 2.0 3.0

Resource density Consumer density

Juveniles Adults

Resource

0 0.2 0.4 0.6 0.8 1.0

0 0.5 1.5 2.0

1.0

Juvenile/adult maintenance ratio (T J/T A)

0 0.1 0.2 0.3 0.4 0.5

Mortality/maintenance ratio (μ/T_A )

✏P f⁰( ¯R)/T_A² = 0

✏P f⁰( ¯R)/T_A² = 5

P = ✏ = T_A = 1, µ = 0.2, T_J = 2.5

Maintenance and stage structure together

§ In equilibrium either juvenile or adult consumer density can increase with increasing mortality (stage-independent,

juvenile, adult)

§ Population cycles can occur as a consequence of stage- structured dynamics

8>

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: dR

dt = P f(R) (C_J + C_A) dC_J

dt = ( f(R) T)⁺ C_A ( f(R) T)⁺ C_J µ C_J dC_A

dt = ( f(R) T)⁺ C_J µ C_A

Maintenance and stage structure together

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1.0

0 0.2 0.4 0.8 0.6 1.0

Mortality rate μ

Juv enile - A dult c onsumer densit y

Adults

P = = T = 1, = 0.75 P = = T = 1, = 1.5

Maintenance and stage structure together

1.0

0 0.2 0.4 0.6 0.8

0 1.0 2.0 3.0

Mortality/maintenance ratio (μ/T)

Adult/juvenile efficiency ratio (β/γ)

Adult density

overcompensation

Juvenile density overcompensation

Maintenance and stage structure together

Resource density

Assimilation & Maturation rate Maintenance

costs

A₀+∆A A₀

M₀+∆M M₀

Assimilation

Maturation

Assimilation & Reproduction rate

Maintenance costs A₀+∆A

A₀

R₀+∆R R₀

Resource density Assimilation

Reproduction

Assimilation increase ≈ 15%

Maturation increase ≈ 35% Assimilation increase ≈ 15%

Reproduction increase ≈ 90%

Net increase in reproduction >> Net increase in maturation

Maintenance and stage structure together

0 20 40 60 80 100

Time

1

0 0.2 0.4 0.6 0.8 1.2

Resource-consumer density

Juveniles Adults

Resource

0 0.1 0.2 0.3 0.4 0.5 0.6

Mortality/maintenance ratio (μ/T)

0 1.0 2.0 3.0

Adult/juvenile assimilation ratio (β/γ) ^{P f0( ¯R)/T2 = 0P f0( ¯R)/T2 = 5}

P = = T = 1, = 2.5

Flies

Jellyfish

Butterflies

Frogs

Development predominates!

Individual life cycles are complex

Flies

Jellyfish

Butterflies

Frogs

Development predominates!

Individual life cycles are complex

Life history’s most prominent feature:

Growth in body size (a doubling at least )

1 mm

x 10⁹

Intra-specific variation in body size!

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

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

Food-dependent growth is ubiquitous

0.01 1 100 10000

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Newborn / adult size ratio

Body mass (g)

Insects Crustaceans Fish Amphibians Reptiles Birds Mammals

~ 1.3 million animal species grow substantially after birth

Source: IUCN List of Threatened Species 2014 De Roos & Persson, MPB 51, 2013

The simplest, fully size-structured analogue

§ Juveniles grow in body size, adults only reproduce

§ Ingestion, maintenance, somatic growth and reproduction proportional to body size

§ Mortality constant within each stage

§ Bio-energetics model: mass conservation

Juveniles and adults may differ in:

• Mass-specific growth and reproduction (net-production rate of new biomass)

• Mortality: µ

^, µ

ν

^{( R),} ν

^{( R)}

The size-structured population model

Size (s)

Density

Juveniles Adults

s_b s_m

Resource

µ_J

µ_A

ω_A^(R)s_m ω_J^(R)s

ν_J^(R)s ν_A^(R)s_m ^s_b

The size-structured population model

Mass conservation:

Juvenile growth and adult reproduction proportional to body size:

g(s, R) = ⌫_J(R)s = ( !_J(R) T)s b(s_m, R) = ⌫_A(R)s_m

s_b = ( !_A(R) T)s_m s_b

@c(t, s)

@t + ⌫_J(R)@(sc(t, s))

@s = µ_Jc(t, s) for s_b  s < s_m

⌫_J(R)s_b c(t, s_b) = ⌫_A(R)s_m

s_b C_A(t) dC_A

dt = ⌫_J(R)s_m c(t, s_m) µ_A C_A(t) dR

dt = ⇢(R_max R) !_J(R)

Z sm

sb

sc(t, s)ds !_A(R)s_mC_A(t)

Equilibrium changes with increasing mortality

0.1 1 10

Stage-independent mortality 0

0.5 1 1.5

Juvenile biomass

0.1 1 10

Juvenile mortality 0.1 1 10 100

Adult mortality

0 1 2 3

Adult biomass

Juvenile biomass Adult biomass

Adult biomass

3

2

1

0

Adult biomass

3

2

1

0

⌫_J( ˜R) > ⌫_A( ˜R) > 0 Reproduction control:

Asymmetric changes in reproduction and maturation with increasing mortality

0.1 1 10

Juvenile mortality 0

0.2 0.4 0.6 0.8

Maturation rate

0 0.4 0.8 1.2 1.6

Reproduction rate

When adults compete more:

§ Low adult fecundity, high juvenile survival

§ Adults dominate

§ Adults use most of their intake for maintenance (no production)

§ Mortality releases adult competition, increases reproduction and juvenile biomass

Biomass maturation rate Biomass reproduction rate Mortality rate

Interplay between maintenance and mortality

Resource

Functional response

Fecundity Maintenance

20% decrease in density increases ingestion by roughly 20%, but doubles adult fecundity

60% increase in total reproduction

Mortality decreases maintenance losses and increases production efficiency

Rate

Equilibrium changes with increasing mortality

0.1 1 10

Stage-independent mortality 0

0.5 1 1.5

Juvenile biomass

0.1 1 10

Juvenile mortality 0.1 1 10 100

Adult mortality

0 1 2 3

Adult biomass

Juvenile biomass Adult biomass

⌫_J( ˜R) > ⌫_A( ˜R) > 0 Reproduction control:

Ontogenetic asymmetry

1 10 100

Consumer Biomass (mg/L)

Mortality (day^-1)

0.01 0.1 0.5

Juveniles Adults

J A

R

J A

R

1 10 100

Consumer Biomass (mg/L)

Mortality (day^-1)

0.01 0.1 0.5

Juveniles Adults

A J

R

J A

R

Reproduction control Mortality increases

juvenile biomass

Development control Mortality increases

adult biomass

⌫

( ˜ R ) > ⌫

( ˜ R ) > 0 ⌫

( ˜ R ) > ⌫

( ˜ R ) > 0