Results

We usea=30,p_S=0.02, K_S=1andT_h=0.6for the simulation, such that the ratios correspond to those in the main investigation (s. Tab.3.1). The higher value for the discovery rate is necessary in order to gain good statistics over all possible reservoir filling states in an adequate time. The higher rate of attacks can be achieved by placing the prey individuals in a smaller arena.

Fig. B.1 shows the resulting fraction of defended prey as a function of the mean amount of secretion in the prey population for different transferring constantse_S(dashed lines). For comparison, we add the corresponding mean-field approximation (solid line) as defined in Eq.(3.13).

We choose different numbers of runs for each set of parameters to achieve comparable statistics since the number of time steps needed to reach equilibrium differ. We use in (a)2000 runs with10⁴ steps each, in (b)1000runs with10⁴steps each, in (c)10⁴runs with1000steps each, and in (d)10⁵runs with100steps each. In (a), we use twice as many data points as in (b), (c), and (d) (which were 10⁷), because we doubled the number of bins for increased precision in (a) in order to resolve the steeper slope.

The mean-field approximation of the fraction of defended prey D(S)differs most from the result of stochastic simulations when the transferring constant e_S is low (i.e. e_S = 0.05; s. Fig. B.1(a)) or high (i.e. e_S = 0.6;

s. Fig.B.1(d)). In the latter case, the fraction of defended prey shows a stepped increase as a function of the mean amount of secretion per prey individual. However, the mean-field approximation and the stochastic simulation provide similar results for intermediate values of the transferring constante_S(s. Fig.B.1(b), (c)).

0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.2 0.4 0.6 0.8 1.0

Amount of secretion S

FractionofdefendedpreyD(S)

(a)

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Amount of secretion S

FractionofdefendedpreyD(S)

(b)

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Amount of secretion S

FractionofdefendedpreyD(S)

(c)

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Amount of secretion S

FractionofdefendedpreyD(S)

(d)

Figure B.1.: The fraction of defended prey plotted against the mean amount of secretion for a population of10 prey individuals (dashed lines). The parameters of the stochastic simulations are: a=30,p_S =0.02, K_S =1, T_h=0.6. The transferring constant is in (a)e_S =0.05, in (b)e_S =0.2, in (c)e_S =0.4, and in (d)e_S=0.6. In (a) we used100bins in comparison to50bins in (b), (c), and (d). Between the bins, we used linear interpolation. For comparison, we add the corresponding mean-field approximation (solid line) as defined in Eq.(3.13).

Fig.B.2shows the time evolution of the consumption rate per predator ((a), (d), (g), (j), (m)), the prey density ((b), (e), (h), (k), (n)), and the fraction of defended prey ((c), (f), (i), (l), (o)) in response to the regeneration rate p_S. In (d)-(f), (g)-(i), (j)-(l), and (m)-(o), we used the fraction of defended prey D(S)from the stochastic simulation shown in Fig.B.1(a), (b), (c), and (d), respectively. For comparison, we show the results for the mean-field approximation of the fraction of defended preyD(S)in Fig.B.2(a)-(c). The parameters that are not varied in a plot are set according to Tab.3.1.

The results that we receive with the stochastic formulation of the fraction of defended prey do not qualitatively differ from those obtained with the mean-field approximation. The smaller the transferring constante_S, the longer it takes the predator to disarm prey individuals and to thus make them consumable (s. Fig.B.2(d), (g), (j), (m) and (f), (i), (l), (o)). The time until all prey individuals are consumed decreases with increasing transferring constant e_S(s. Fig.B.2(b), (e), (h), (k), (n)).

Although the stepped increase of the fraction of defended prey as a function of the mean amount of secretion per prey individual received for a large transferring constante_S (s. Fig.B.1(d)) transfers to the dynamics of prey consumption (s. Fig.B.2(m)-(o)), the general trends are the same as in the mean-field approximation. Especially the results obtained with the transferring constante_S=0.2(s. Fig.B.2(g)-(i)), which we used in a large part of our study, differ even quantitatively only slightly from those received in the mean-field approximation (s. Fig. B.2(a)-(c)). We thus conclude that our expression for the fraction of defended preyD(S)(s. Eq.(3.13)) is appropriate.

0 20 40 60 80 100 0

1.5 3.

4.5 6.

7.5 9.

Time in h Regen.ratepS

(a) _Functional

response

0.04 0.08 0.12 0.16 0.20 0.24 0.28

0 20 40 60 80 100 0

1.5 3.

4.5 6.

7.5 9.

Time in h Regen.ratepS

(b)

Prey density

0.062 0.124 0.186 0.248 0.310 0.372 0.434

0 20 40 60 80 100 0

1.5 3.

4.5 6.

7.5 9.

Time in h Regen.ratepS

(c) Fraction of

defended prey

0.22 0.33 0.44 0.55 0.66 0.77 0.88

0 20 40 60 80 100 0

1.5 3.

4.5 6.

7.5 9.

Time in h Regen.ratepS

(d) Functional

response

0.049 0.098 0.147 0.196 0.245 0.294 0.343

0 20 40 60 80 100 0

1.5 3.

4.5 6.

7.5 9.

Time in h Regen.ratepS

(e)

Prey density

0.062 0.124 0.186 0.248 0.310 0.372 0.434

0 20 40 60 80 100 0

1.5 3.

4.5 6.

7.5 9.

Time in h Regen.ratepS

(f) Fraction of

defended prey

0.22 0.33 0.44 0.55 0.66 0.77 0.88

0 20 40 60 80 100 0

1.5 3.

4.5 6.

7.5 9.

Time in h Regen.ratepS

(g) _Functional

response

0.036 0.072 0.108 0.144 0.180 0.216 0.252

0 20 40 60 80 100 0

1.5 3.

4.5 6.

7.5 9.

Time in h Regen.ratepS

(h)

Prey density

0.062 0.124 0.186 0.248 0.310 0.372 0.434

0 20 40 60 80 100 0

1.5 3.

4.5 6.

7.5 9.

Time in h Regen.ratepS

(i) Fraction of

defended prey

0.22 0.33 0.44 0.55 0.66 0.77 0.88

0 20 40 60 80 100 0

1.5 3.

4.5 6.

7.5 9.

Time in h Regen.ratepS

(j) _Functional

response

0.034 0.068 0.102 0.136 0.170 0.204 0.238

0 20 40 60 80 100 0

1.5 3.

4.5 6.

7.5 9.

Time in h Regen.ratepS

(k)

Prey density

0.062 0.124 0.186 0.248 0.310 0.372 0.434

0 20 40 60 80 100 0

1.5 3.

4.5 6.

7.5 9.

Time in h Regen.ratepS

(l) Fraction of

defended prey

0.22 0.33 0.44 0.55 0.66 0.77 0.88

0 20 40 60 80 100 0

1.5 3.

4.5 6.

7.5 9.

Time in h Regen.ratepS

(m) _Functional

response

0.04 0.08 0.12 0.16 0.20 0.24 0.28

0 20 40 60 80 100 0

1.5 3.

4.5 6.

7.5 9.

Time in h Regen.ratepS

(n)

Prey density

0.062 0.124 0.186 0.248 0.310 0.372 0.434

0 20 40 60 80 100 0

1.5 3.

4.5 6.

7.5 9.

Time in h Regen.ratepS

(o) Fraction of

defended prey

0.22 0.33 0.44 0.55 0.66 0.77 0.88

Figure B.2.: The dynamics of prey consumption as a function of time and the regeneration ratep_Susing the four functional forms to describe the fraction of defended prey D(S)shown in Fig.B.1(e_S =0.05: (d)-(f);

e_S=0.2: (g)-(i);e_S=0.4: (j)-(l);e_S=0.6: (m)-(o)). For comparison we added the results for the version used in the main part of this thesis ((a)-(c)). The different color shades indicate the prey consumption rate ((a), (d), (g), (j), (m)), prey density ((b), (e), (h), (k), (n)), and the fraction of defended prey ((c), (f), (i), (l), (o)). The parameters that are not varied in a plot are set to the values given in Tab.3.1.

C Trade-off between the reservoir size and the growth rate on the long time scales

Similar to the investigation of reducible defense in the long-term limit (s. Fig. 3.3), we analyze the trade-off between investing in growth and defense via the reservoir sizeK_Son long time scales.

Fig.C.1shows (a) the prey density, (b) the predator density, and (c) the fraction of defended prey in response to the reservoir sizeK_Sand its cost factorϑ. The white region indicates where the predator goes extinct.

As expected, we find the qualitative same behavior as in Fig.3.3, where we varied the investment in defense via the regeneration rate p_S: The prey benefits from defense when the associated costs are not too high. When the costs are low, the predator also benefits from defense as the increase of the consumption rate due to the higher prey density outweighs its decrease caused by the higher fraction of defended prey. The prey can even benefit from increasing costs for defense since the decreased consumption rate due to the decreased predator density outweighs this cost increase (i.e. for K_S =75and an increasing cost factorϑ >0.0015). This is concert with our results in Section3.6.

Furthermore, both predator and prey first suffer from a small, but increasing reservoir sizeK_Sfor a high and fixed cost factorϑas illustrated by the color change from lighter to darker color with increasingK_S. This is the case as the costs for defense are higher than the benefit which leads to the decrease in prey density and as a consequence to the decrease in predator density. The prey density then increases for further increasing reservoir sizeK_S. In contrast to the findings of Fig. 3.3, the predator density also increases and then decreases again with further increasing investment in defense via the reservoir sizeK_S. This indicates that investment in defense becomes beneficial for the prey due to the decrease in predator density, although this decrease is non-monotonical.

0 50 100 150 0.000

0.002 0.004 0.006 0.008

Co st fa cto r ϑ

0 50 100 150

0.000 0.002 0.004 0.006 0.008

Co st fa cto r ϑ

0 50 100 150

Reservoir size K

S

0.000 0.002 0.004 0.006 0.008

Co st fa cto r ϑ

0.0034 0.0048 0.0062 0.0076 0.0090 0.0104

Prey density

0.00870 0.02194 0.03518 0.04842 0.06166 0.07490

Predator density

0.0000 0.1444 0.2888 0.4333 0.5777 0.7221

Fraction of defended prey

Figure C.1.:(a) Prey density, (b) predator density, and (c) the fraction of defended prey displayed in color code as a function of reservoir sizeK_Sand its cost factorϑ. The white region indicates where the predator goes extinct.

D Stochastic migration

We use the Gillespie algorithm (Gillespie,1976,1977) for the implementation of stochastic migration, which we explain in detail in this section.

D.1 Implementation via Gillespie algorithm

For stochastic migration, we follow the following plan of procedures:

1. Determinate the next migration event. This includes the choice of the directed link, the migrating species as well as the time of the next migration event (the exact calculations are shown below in Eq.(D.5)).

2. Calculate local population dynamics until the time of the next migration event is reached.

3. When the time of the next migration event is reached shift the migrating biomass unitB_migrto the determined patch.

4. Go back to step one until the set simulation time has ended.

As in Chapter4, we denote the number of species asSand the number of patches asY. The patchµfor emigration is chosen proportionally to the amount of biomass that the patch contains, i.e. B^µ=PS

i=1B_i^µ, and the number of links that the patch has which we denote ash^µ. The patchµis chosen for an emigration event with a rate

a^µ=dh^µ B^µ

B_migr, (D.1)

with migration strength d (cp. Eq.(4.7)) and the migrating biomass unit B_migr. The patch of immigration is then chosen randomly among all patches that are connected to the patch of emigration, i.e. with probability _h¹_µ. Hence, the probability that patchµis chosen for immigration depends on the probability that a connected patchνis chosen for emigration, i.e.

e^µ=X

ν∈P_µ

a^ν

h^νa₀, (D.2)

whereP_µis the set of patches linked to the patchµand

a₀=

Y

X

µ=1

a^µ(D.1=⁾

Y

X

µ=1

dh^µ B^µ

B_migr. (D.3)

Note, thatP

µe^µ=1. The selection probability for a species to migrate is chosen proportionally to its biomass in the patch, i.e.

s^µ_i = B^µ_i

B^µ. (D.4)

In practice, the time of the next migration eventτ, as well as the patchµfor emigration and the speciesξinvolved in the migration event are calculated via the random numbers r₁, r₂, and r₃, that are chosen uniformly from an interval(0, 1), as follows

τ= 1 a₀ln

1 r₁

‹

µ−1

X

i=1

aⁱ<r₂a₀¶ Xµ

i=1

aⁱ

ξ−1

X

i=1

s^u_i <r₃s₀¶ Xξ

i=1

s^u_i ,

(D.5)

withs₀=PS i=1s_i.

Indeed, with this choice of parameter values, we end up with the same equation for migration as in the deter-ministic limit (s. Eq.(4.7)). We will show this in the following.

Let us consider Z migration events in the whole simulation timeT. In each migration event a fixed migrating biomass unitB_migr migrates. Hence, the amount of migrating biomass per time step is ^{Z Bmigr}_T . As the number of migration events isZ= ^T_τ¯ with the mean time between two migration eventsτ¯, the amount of migrating biomass per time step is

M= B_migr

τ¯ . (D.6)

In order to describe the migrating biomass per species, time step, and patch as in the deterministic case in Eq.(4.7), we need to include the probability that the patchuis chosen for immigration or emigration as well as the probability that speciesimigrates as defined above (i.e. Eq.(D.1), (D.2), and (D.4)). We thus find

M_i^u(B) =~ B_migr τ¯

X

v∈Pu

a^v

a₀h^vs^v_i −B_migr τ¯

a^u

a₀s^u_i . (D.7)

Using the time between two migration eventsτ(s. Eq.(D.5)), we find for its mean

τ¯= 1 a₀

(D.3=)

‚ dX

i,u

B^u_i B_migrh^u

Œ−1

. (D.8)

In SectionD.2, we show the first step in detail. By including this expression (D.8) in Eq.(D.7) (in form ofτ¯=_a¹₀), we find

M_i^u(B~) =B_migrX

v∈P_u

a^v

h^vs^v_i −B_migra^us^u_i

(D.1)

= dX

v∈Pu

‚ X

i

B_i^v

Œ s_i^v−d

‚ X

i

h^uB^u_i

Πs^u_i

(D.4)

= dX

v∈Pu

‚ X

i

B_i^v

ΠB^v_i P

iB^v_i −dh^u

‚ X

i

B^u_i

ΠB_i^u P

iB^u_i

=d X

v∈Pu

B^v_i −h^uB_i^u

!

=dX

v∈Pu

B^v_i −B_i^u

. (D.9)

Hence, considering a large number of individuals that migrate we obtain the same migration term as in the deter-ministic case (cp. Eq.(4.7)).

D.2 Mean time between two migration events

Here, we concentrate on the calculations of the mean time between two migration events, i.e. τ¯ = _a¹₀. The time between two migration events isτ= _a¹₀ln€₁

r₁

Š with a random number r₁which is uniformly distributed in r₁∈(0, 1). We will thus show that the expected valueE€

ln€₁

r₁

ŠŠ=1.

In our case, the expected value can be calculated as follows:

E(g(X)):=

Z

g(z)fX(z)dz (D.10)

with f_X(z) =

¨1 0<z<1

0 otherwise andg(z) =ln ¹_z

. Hence, we find

limc→0

Z1 c

ln

1 z

‹

dz=−lim

c→0

Z1 c

ln(z)dz

=−lim

c→0[zln(z)−z]¹_c

=−lim

c→0(−1−(cln(c)−c)) =1. (D.11) Thus, we have shown thatτ¯= _a¹₀, when the time between two migration events isτ= _a¹₀ln€₁

r₁

Šwith a random numberr₁∈(0, 1).

E Generation based modeling

In Chapter5, we describe the population dynamics on generation basis, since the investigated main features, i.e.

plant trait variability and herbivore preference, usually cannot be adapted on the typical time scale of feeding interactions. We rather assume that changes of these features are based on plastic responses to the environment or changes in the genotype of the species. These features can hence be seen as evolutionary strategies. The basic population dynamics equation shown in Eq.(2.1) can easily be transformed to a generation-based equation, which we will show in the following.

In order to distill theper seeffect of plant trait variability on the herbivore population, we focus on the herbivore population dynamics. Hence, we assume that plant population is large enough to be considered as constant over the time scales covered in our model. Consequently, the herbivore population is small compared to the plant population, such that we assume that intraspecific competition is negligible. Starting with the predator dynamics of our basic consumer-resource model defined in Eq.(2.1), we find by neglecting intraspecific competition and by integrating over the time of one generation[t₁,t₂]

dC

dt = (λF(R)−αC)C ZC(t₂)

C(t1)

dC C =

Z t₂ t₁

(λF(R)−αC)dt C(t2)

C(t₁) =exp((λF(R)−αC) (t₂−t₁)) C(τ+1) =C(τ)exp(λF(R)−αC)

C(τ+1) =C(τ)W (E.1)

with the mean population fitness

W =exp(λF(R)−αH)

=ΛA. (E.2)

Here,Λdescribes the feeding interaction andAthe metabolic loss per predator during the time of one generation.

We are, however, interested in intraspecific variability in a resource trait, such that we choose another way to describe the mean fitness. We introduce this model in Section5.3.

F Proof Jensen’s inequality

In this section, we proof Jensen’s inequality, which states that

f

‚ _n X

i=0

λiz_i

Œ

≤

n

X

i=0

λif(z_i) (F.1)

withPn

i=0λi=1,n∈N, when f(z)is a concave upwards function. A function is concave upwards when

f(t x+ (1−t)y)≤t f(x) + (1−t)f(y), (F.2) for allt∈(0, 1). We use induction to proof this theorem:

• Base case:Ben=0; thenλ0=1and we find that

f (λ0z₀) =f (z₀) =λ0f(z₀). (F.3)

• Induction assumption:We assume that Eq.(F.1) is true for ann∈N.

• Induction step:Let’s considern+1, i.e

n+1

X

i=0

λi=1

λ_n+1=1− Xn

i=0

λi. (F.4)

We find

f

‚_n+1 X

i=0

λiz_i

Œ

=f

‚ _n X

i=0

λiz_i+λn+1z_n+1

Œ

=f











(1−λn+1) Xn

i=0

λiz_i (1−λn+1)

| {z }

z⁰

+λn+1z_n+1











=f (1−λn+1)z⁰+λn+1z_n+1

(F.2)

≤ (1−λn+1)f(z⁰) +λn+1f(z_n+1)

= (1−λn+1)f











n

X

i=0

λi

(1−λn+1)

| {z }

λ⁰_i

z_i











+λn+1f(zn+1)

= (1−λn+1)f

‚ _n X

i=0

λ⁰_iz_i

Œ

+λn+1f(z_n+1).

Here, we use the induction assumption and find

f

‚_n+1 X

i=0

λiz_i

Œ

≤(1−λ_n+1)

n

X

i=0

λ⁰_if (z_i) +λ_n+1f(z_n+1)

=

n

X

i=0

λif (zi) +λn+1f(zn+1)

=

n+1

X

i=0

λif (z_i). (F.5)

Hence, we showed the validity of Eq.(F.1) for alln∈N.

G Robustness tests in the plant-herbivore model

In this chapter, we test the robustness of the results of Chapter5in response to changes of parameters and consid-ered functions in the model.

G.1 Proportional growth deficiency

First, we test the robustness of our results under changes in the functional form of the proportional growth defi-ciencyg(d,ν)(cp. Eq.(5.37)). Another function that satisfies the assumptions in Section5.5.1with an appropriate choice of parameter values, is

g(d,ν) =max

1− ν³

8500dexp(d/ν), 0

. (G.1)

Fig. G.1 shows the resulting performance functions and the fitness W_H(d) of a herbivore individual feeding on a leaf with defense level d in dependency of the predator encounter rate a₀ (cp. Eq.(5.39)) and the correlation parameterl (cp. Eq.(5.32)). We use the parameter values shown in Tab.G.1for this investigation. The functions have a similar form as in Fig.5.13.

Table G.1.:Parameter values used for distinguishing the different herbivore strategies using the proportional growth deficiencyg(d,ν)of Eq.(G.1).

Generalist Non-seq. specialist Seq. specialist

Cost parameterν 12.5 2.5 0.67

Generalist benefit factorγ 2 1 1

Efficiency of converting defenseθS 0 0 10

Fig. G.2 shows the mean fitness of a herbivore population consisting of generalists, sequestering, and non-sequestering specialists in dependency of the plant strategy parameterS(cp. Eq.(5.5)) and herbivore preference τ (cp. Eq.(5.9)) for different correlation parameter l. We used a predator encounter rate of a₀ =0.25 for this investigation. The blue line indicates the optimal herbivore preference, i.e. the preference that maximizes herbivore fitness for a given plant strategy parameter S. Again, the white region shows the conditions under which herbivore fitness is below one which means that the herbivore population goes extinct with proceeding time.

The fitness landscapes do not qualitatively differ from those in Fig.5.15. As the sequestering specialist can deal worse with plant defense with our choice of parameter values than in Section5.5, the fitness values change in a broader range. Hence, the fitness increase with decreasing plant strategy parameterSwhen approximatelyS>0.9 (as herbivores show no preference and have a concave downwards performance function) is more clearly visible.

We conclude that the results are robust under changes of the functional form of the proportional growth deficiency g(d,ν)under the assumptions proposed in Section5.5.1.

0 1 2 3 4 5 6

Fit ne ss W

H

( d )

(a) l =0 . 0

a

0

=0 . 25

NsS SS Gen

(b) a

₀

=0 . 25 l =0 . 0

NsS SS Gen

0 1 2 3 4 5 6

Fit ne ss W

H

( d )

(c) l =0 . 5 NsS

SS Gen

(d) a

₀

=0 . 50 NsS

SS Gen

0 2 4 6 8 10

Defense level d 0

1 2 3 4 5 6

Fit ne ss W

H

( d )

(e) l =1 . 0 NsS

SS Gen

0 2 4 6 8 10

Defense level d

(f) a

₀

=0 . 75 NsS

SS Gen

0 100 200 300 400 500 600

Pe rfo rm an ce f ( d )

Figure G.1.:FitnessW_H(d)(cp. Eq.(5.38)) of a herbivore individual that feeds on a leaf with defense level d us-ing the proportional growth deficiencyg(d,ν)of Eq.(G.1) under the assumption of different predator encounter ratesa₀(cp. Eq.(5.39)) and correlation parametersl(cp. Eq.(5.32)).

0.2 0.4 0.6 0.8 0.2

0.4 0.6 0.8

Generalist (a)

l=0.0

0.2 0.4 0.6 0.8 0.2

0.4 0.6 0.8

N.-s. Specialist (b)

0.2 0.4 0.6 0.8 0.2

0.4 0.6 0.8

Sequ. Specialist (c)

0.2 0.4 0.6 0.8 0.2

0.4 0.6 0.8

Plant strategy S

(d)

l=0.5

0.2 0.4 0.6 0.8 0.2

0.4 0.6

0.8 (e)

0.2 0.4 0.6 0.8 0.2

0.4 0.6

0.8 (f)

0.2 0.4 0.6 0.8 Preference τ 0.2

0.4 0.6

0.8 (g)

l=1.0

0.2 0.4 0.6 0.8 Preference τ 0.2

0.4 0.6

0.8 (h)

0.2 0.4 0.6 0.8 Preference τ 0.2

0.4 0.6

0.8 (i)

1.000 1.257 1.515 1.772 2.030 2.287 2.545 2.802

Fitness

Figure G.2.:Mean fitness (i.e. the mean number of offspring per herbivore individual reaching reproductive age;

cp. Eq.(5.41)) of a population of generalists, sequestering, and non-sequestering specialists as a function of herbivore preferenceτ(cp. Eq.(5.9)) and the plant strategy parameterS(cp. Eq.(5.5)). We used the proportional growth deficiency g(d,ν)of Eq.(G.1) for this figure. The correlation parameter of the nutrient and defense level in the leavesl(and thus the nutrient level variability; cp. Eq.(5.32)) increases from top to bottom row. The blue line indicates the optimal herbivore preference for a given plant strategy parameterS, i.e. the preferenceτfor which herbivore fitness is maximized.

G.2 Probability to be consumed by a predator

Here, we test the robustness of our results under changes of the functional form of the probability to be consumed by a predatora(d)(cp. Eq.(5.39)). Hence, we focus on sequestering specialists in this section. Other functions that satisfy the assumptions in Section5.5.1with an appropriate choice of the parameter values, are

a₁(d) =a₀e^−0.5θd, (G.2)

a₂(d) = a₀

1+θd³, (G.3)

and are shown in Fig.G.3. We denote the functional form that we used in the main part of this thesis asa₀(d)in this section (cp. Eq.(5.39)). For the efficiency of converting plant defense, we useθS=10for all functional forms of the probability to be consumed by a predatora(d). We analyze the impact of varying values of the efficiency of converting plant defenseθSin SectionG.3.

0 2 4 6 8 10

Defense level d 0.00

0.05 0.10 0.15 0.20 0.25

Pr ob ab ilit y t o b e c on su m ed a ( d ) a

0

( d ) a

1

( d ) a

2

( d )

Figure G.3.:Different functional forms of the probability to be consumed by a predatora(d)fora₀=0.25:a₀(d) =

1+θd1 (cp. Eq.(5.39)),a₁(d) =e^−0.5θd, anda₂(d) =_1+θ¹_d3. We usedθS=10for all considered functions.

The different functional forms of the probability to be consumed by a predatora(d)only have a small impact on the fitnessW_H(d)(cp. Eq.(5.38)) of a sequestering specialist individual that feeds on a leaf with defense leveld (s. Fig.G.4).

Fig.G.5shows the mean fitness of a herbivore population consisting of sequestering specialists in dependency of the plant strategy parameter S (cp. Eq.(5.5)) and herbivore preference τ (cp. Eq.(5.9)) for these different functional forms of the probability to be consumed by a predatora(d). The predator encounter ratea₀ increases from the left to the right panels. For this investigation, we assume that all leaves contain the same nutrient level, i.e. l = 0 (cp. Eq.(5.32)). The blue line indicates the optimal herbivore preference, i.e. the preference that maximizes herbivore fitness for a given plant strategy parameterS. The white region shows the conditions under which herbivore fitness is below one which means that the herbivore population goes extinct with proceeding time.

The fitness landscapes do not differ qualitatively. The predator encounter ratea₀for which a herbivore popula-tion that feeds on a plant withS>0is fitter than one feeding on a plant withS=0differ slightly between the different functional forms of the probability to be consumed by a predatora(d).

We conclude that the results are robust under changes of the functional form of the probability to be consumed by a predatora(d).

0 2 4 6 8 10 0

1 2 3 Fit ne ss W

H

( d )

(a)

a

0

=0 . 25 a

0

( d ⁾

a

1

( d ⁾ a

2

( d ⁾

0 2 4 6 8 10

0 1 2 3 Fit ne ss W

H

( d )

(b)

a

0

=0 . 50

0 2 4 6 8 10

Defense level d 0

1 2 3 Fit ne ss W

H

( d )

(c)

a

0

=0 . 75

Figure G.4.:FitnessW_H(d)(cp. Eq.(5.38)) of a sequestering specialist individual that feeds on a leaf with defense level d assuming different functional forms of the probability to be consumed by a predator a(d) (cp. Fig. (G.3)). From up to down, we increase the predator encounter ratea₀.

0.2 0.4 0.6 0.8 0.2

0.4 0.6 0.8

Plant strategy S

a0(d)

(a) a₀=0.25

0.2 0.4 0.6 0.8 0.2

0.4 0.6

0.8 (b)

a₀=0.50

0.2 0.4 0.6 0.8 0.2

0.4 0.6

0.8 (c)

a₀=0.75

0.2 0.4 0.6 0.8 0.2

0.4 0.6 0.8

Plant strategy S

a1(d)

(d)

0.2 0.4 0.6 0.8 0.2

0.4 0.6

0.8 (e)

0.2 0.4 0.6 0.8 0.2

0.4 0.6

0.8 (f)

0.2 0.4 0.6 0.8 Preference τ 0.2

0.4 0.6 0.8

Plant strategy S

a2(d)

(g)

0.2 0.4 0.6 0.8 Preference τ 0.2

0.4 0.6

0.8 (h)

0.2 0.4 0.6 0.8 Preference τ 0.2

0.4 0.6

0.8 (i)

1.000 1.214 1.427 1.641 1.854 2.068 2.281 2.495 2.708

Fitness

Figure G.5.:Mean fitness (cp. Eq.(5.41)) of a sequestering specialist population in response to the plant strategy parameterS(cp. Eq.(5.5)) and herbivore preferenceτ(cp. Eq.(5.9)) assuming different functional forms of the probability to be consumed by a predatora(d)(cp. Fig. (G.3)). The predator encounter ratea₀ increases from the left to the right panels.

G.3 Efficiency of converting plant defense

Here, we test the robustness of the fitness landscape of the sequestering specialist in response to changes of the efficiency of converting plant defenseθS(cp. Eq.(5.39)).

Fig.G.6shows the fitnessW_H(d)(cp. Eq.(5.38)) of a sequestering specialist individual that feeds on a leaf with defense level d for different values of the efficiency of converting plant defense θS (cp. Eq.(5.39)). The higher the efficiency of converting plant defenseθS, the higher is the benefit of the sequestering specialist that feeds on weakly or medium-defended leaves.

0 2 4 6 8 10

Defense level d 0.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5

Fit ne ss W

H

( d )

θ

_S

=0 . 1 θ

S

=1 θ

S

=10 θ

_S

=100

Figure G.6.:FitnessW_H(d)(cp. Eq.(5.38)) of a sequestering specialist individual that feeds on a leaf with defense leveldassuming different values of the efficiency of converting plant defenseθS(cp. Eq.(5.39)).

Fig. G.7 shows the mean fitness of a sequestering specialist population for different values of the efficiency of converting plant defenseθS (cp. Eq.(5.39)) and in response to the plant strategy parameter S and herbivore preferenceτ. In the left panels, we use a predator encounter rate ofa₀=0.25, in the right panels,a₀=0.75.

The efficiency of converting plant defenseθ_Sonly has a slight quantitative impact on the fitness landscape of the sequestering specialist. The reason is that the fitnessW_H(d)(cp. Eq.(5.38)) of a sequestering specialist individual that feeds on a leaf with defense leveld only changes severely for weakly defended leaves, but as is it not worth to show high preference, these changes have only a slight impact on the fitness landscape. Consequently, the mean fitness changes most when the plant strategy parameterSis low as preference has, in this case, the largest impact (cp. Fig.5.2(c), (d)). As the efficiency of converting plant defenseθSonly changes the fitnessW_H(d)(cp. Eq.(5.38)) of a sequestering specialist individual that feeds on a leaf with defense level d quantitatively, the mean fitness is also only quantitatively affected.

We conclude that the results are robust under changes of the efficiency of converting plant defenseθS.

0.2 0.4 0.6 0.8 0.2

0.4 0.6 0.8

Plant strategy SθS=0.10

(a) a₀=0.25

0.2 0.4 0.6 0.8 0.2

0.4 0.6

0.8 (b)

a₀=0.75

0.2 0.4 0.6 0.8 0.2

0.4 0.6 0.8

Plant strategy SθS=1.00

(c)

0.2 0.4 0.6 0.8 0.2

0.4 0.6

0.8 (d)

0.2 0.4 0.6 0.8 0.2

0.4 0.6 0.8

Plant strategy SθS=10.00

(e)

0.2 0.4 0.6 0.8 0.2

0.4 0.6

0.8 (f)

0.2 0.4 0.6 0.8 Preference τ 0.2

0.4 0.6 0.8

Plant strategy SθS=100.00

(g)

0.2 0.4 0.6 0.8 Preference τ 0.2

0.4 0.6

0.8 (h)

1.000 1.240 1.480 1.720 1.959 2.199 2.439 2.679 2.919

Fitness

Figure G.7.:Mean fitness (cp. Eq.(5.41)) of a sequestering specialist population in response to the plant strategy parameterS(cp. Eq.(5.5)) and herbivore preferenceτ(cp. Eq.(5.9)) assuming different values of the efficiency of converting plant defenseθS(cp. Eq.(5.39)). In the left panels, we use a predator encounter rate ofa₀=0.25, in the right panels,a₀=0.75.

H Coexistence of plants and herbivores

When the maximal herbivore population, that can coexist, is limited by the available plant biomass because of limited food or place for oviposition, we expect that the herbivore may benefit from other situations. For instance, the benefit of a higher plant biomass may outweigh the fitness loss due to unfavorable conditions for the herbivore.

In order to investigate this, we use a similar approach as in (Gaschler,2019). Indeed, the extensions to the basic model (s. Section5.3) used in this section were developed by Sarah Gaschler and Barbara Drossel. Furthermore, Sarah Gaschler found the parameter values that are based on empirical values. The author of this thesis applied this model to investigate the impacts of the plant strategy parameterS and herbivore preferenceτon herbivore density and plant biomass.

We assume a logistic growth of the plant biomass Mwith a growth rate r and a carrying capacityK_P. Further-more, the plant suffers loss due to herbivory with a herbivore feeding rate a. The herbivore population grows according to its mean fitnessW_H, but is limited by the plant biomassM(t). In terms of equations, this means

M(t+1) =M(t) +r M(t)

1− M(t) K_P

‹

−aH(t), H(t+1) =W_HH(t)e^{−H(t)/(KHM(t))},

(H.1)

where the herbivore limitation factorK_H limits herbivore density per plant biomass.

The parameter values that we used for the growth raterand the carrying capacity of plant biomassK_Pare listed in Tab.H.1and are motivated by empirical values (Gaschler,2019). Furthermore, we use the concave downwards performance functionf_negshown in Fig.5.4for this investigation. When the herbivore population is not limited by plant biomass, we would thus expect that a herbivore population with low preference benefits from low nutrient level variability, but suffers from this situation when having strong preference (cp. Fig.5.6(c)).

Table H.1.: Parameters used to model the coexistence of plant and herbivore.

Growth rate Carrying capacity of plant biomass

r K_P

1.25 100

Fig.H.1 shows the plant biomass and the herbivore population at the stable fixed point in dependency of the plant strategy parameterSand herbivore preferenceτ. From the top to the bottom row, we increase the herbivore limitation factorK_H.

When the herbivore limitation factorK_H is small (cp. Fig.H.1(a), (b)), the herbivore population shows a similar behavior as in Fig.5.6(c) – a herbivore population that has low preference benefits from low nutrient level vari-ability, but suffers from this situation when having strong preference. The plant mass shows the contrary trend, i.e. the plant mass increases when the herbivore population decreases. In this case the plant mass is high since the herbivore population is small due to the low herbivore limitation factorK_H. Consequently, the herbivore is not limited by plant mass.

However, the higher the herbivore limitation factorK_H, and thus the higher the plant loss due to herbivory, the higher is the herbivore benefit of a plant strategy parameterSand a herbivore preferenceτthat allow high plant biomasses. This means that both plant and herbivore population benefit from high nutrient level variability (i.e.

lowS) when the herbivore has low preference as illustrated by the color change from lighter to darker color with decreasingS. The opposite is true when the herbivore has high preference. The decreased loss due to herbivore limitation thus outweighs the fitness decrease due to less favorable conditions for the herbivore.

0.1 0.2 0.3 0.4 0.0

0.2 0.4 0.6 0.8 1.0

Preferenceτ

PlantstrategyS

(a)

Herbivore density

64.8 68.4 72.0 75.6 79.2 82.8 86.4 90.0

0.1 0.2 0.3 0.4 0.0

0.2 0.4 0.6 0.8 1.0

Preferenceτ

PlantstrategyS

(b)

Plant mass

76.56 77.44 78.32 79.20 80.08 80.96 81.84 82.72 83.60 84.48

0.1 0.2 0.3 0.4 0.0

0.2 0.4 0.6 0.8 1.0

Preferenceτ

PlantstrategyS

(c)

Herbivore density

117.00 118.04 119.08 120.12 121.16 122.20 123.24 124.28

0.1 0.2 0.3 0.4 0.0

0.2 0.4 0.6 0.8 1.0

Preferenceτ

PlantstrategyS

(d)

Plant mass

41.8 44.0 46.2 48.4 50.6 52.8 55.0 57.2 59.4 61.6

0.1 0.2 0.3 0.4 0.0

0.2 0.4 0.6 0.8 1.0

Preferenceτ

PlantstrategyS

(e)

Herbivore density

98.6 102.0 105.4 108.8 112.2 115.6 119.0 122.4

0.1 0.2 0.3 0.4 0.0

0.2 0.4 0.6 0.8 1.0

Preferenceτ

PlantstrategyS

(f)

Plant mass

28.6 31.2 33.8 36.4 39.0 41.6 44.2 46.8 49.4 52.0

Figure H.1.:Herbivore population and plant biomass on the stable fixed point in dependency of the plant strategy parameterSand herbivore preferenceτ. The panels differ in the used herbivore limitation factorK_H. In (a), (b) we usedK_H=1, in (c), (d)K_H=2.5, and in (e), (f)K_H =3.

Curriculum Vitae

2016 - 2019Ph.D. in the group of B. Drossel, Physics Technische Universität Darmstadt Migrate, defend, and evolve: Theoretical ecology presented in three different ways

2014 - 2016Master of Science, Physics Technische Universität Darmstadt

Influence of Stochastic Migration on Spatial Food Webs in the Niche Model

2012 - 2013Study abroad in Lausanne, Switzerland École polytechnique fédérale Lausanne

2010 - 2014Bachelor of Science, Physics Technische Universität Darmstadt

Molekulardynamische Simulationen von Elastin in Confinements unterschiedlicher Geometrie

2001 - 2010Allgemeine Hochschulreife Karl-Rehbein-Gymnasium Hanau

Teaching experience at TU Darmstadt

Supervised master thesis

2018 - 2019Sarah Gaschler (Gaschler,2019)

Pflanzenverteidigung – Effektive Reduktion von Herbivorie auf kurzen und langen Zeitskalen

Supervised bachelor theses 2017 Kenneth van Beek

Allometrische Skalierung in Nahrungsnetzen mit zwei trophischen Niveaus 2017 Esra Bauer

Skalengesetze in Räuber-Beute-Systemen

2015 - 2016Christian Hoch (Hoch,2016)

Robustheit von Nahrungsnetzen im AFK- und Nischenmodell

Teaching

WS 18/19 Supervisor of a tutorial class (and lecture stand-in) AG Drossel

Komplexe dynamische Systeme

SS 18 Seminar assistant AG Drossel

Statistische Physik von Netzwerken

WS 17/18 Supervisor of a tutorial class (and lecture stand-in) AG Drossel

Statistische Physik

SS 17 Tutorial coordinator AG Feile

Physik für Bauingenieure

SS 17 Lecture assisistent for experiments AG Stühn

Experimentalphysik I

WS 16/17 Lecture assistant for experiments ^{AG Vogel}

Physik I für Chemiker/MaWi

Im Dokument Migrate, defend, and evolve: Theoretical ecology presented in three different ways (Seite 105-129)