Reversible phenotypic plasticity with continuous adaptation

Abstract

We introduce a novel model for continuous reversible phenotypic plasticity. The model includes a one-dimensional environmental gradient, and we describe performance of an organism as a function of the environmental state by a Gaussian tolerance curve. Organisms are assumed to adapt their tolerance curve after a change of the environmental state. We present a general framework for calculating the genotype fitness if such adaptations happen in a continuous manner and apply the model to a periodically changing environment. Significant differences of our model with previous models for plasticity are the continuity of adaptation, the presence of intermediate phenotypes, that the duration of transformations depends on their extent, fewer restrictions on the distribution of the environment, and a higher robustness with respect to assumptions about environmental fluctuations. Further, we show that continuous reversible plasticity is beneficial mainly when environmental changes occur slow enough so that fully developed phenotypes can be exhibited. Finally we discuss how the model framework can be generalized to a wide variety of biological scenarios from areas that include population dynamics, evolution of environmental tolerance and physiology.

Appendix A

1.1 A.1 Calculating the fitness of a continuously reversibly plastic genotype

In this appendix we derive a formula for the fitness of a genotype in the scenario introduced in Sects. 2 and 3. That includes that the peak $(m,\frac{1}{\sqrt{2 \pi } b})$ of a normalized tolerance curve shifts with constant speed $v_1$ (respectively $v_0$) towards the peak of an induced (respectively non-induced) tolerance curve in order to adjust the phenotype to a periodically changing environment. A stress-free environmental state is interrupted by stress events with frequency $p$ and the stress events are assumed to have length $t$ and their stress intensities are randomly distributed with mean $\overline{\phi }$ and standard deviation $\sigma _\phi$. Individuals are born at uniformly distributed times and we will use the law of large numbers to average out the stochasticity in the individual lives and calculate the fitness of the genotype. We then maximize the fitness numerically by finding optimal parameter values for the induced and non-induced tolerance curve. To increase comprehensibility of this section, the reader is referred to a list of notations and abbreviations at the end of this appendix.

Fig. 7

Change of the phenotypic state $s$ over time. The striped boxes symbolise stress events and the lines in between are stress-free times. The upper course describes lives that begin during a stress event, and the lower course describes lives that begin during a stress-free time. Dashed lines indicate phases that are only partly experienced by a given individual, depending on the exact time of birth, whereas solid lines indicate phases that are experienced completely by all individuals, depending only on whether their life starts during stress or not

The following calculations base on the fact that—due to the assumption that phenotypes are fully developed at birth—all organisms born during stress (and likewise all organisms born during stress-free times) have a common course of the phenotypic state for part of their lives given the same sequence of stress events and given the stress frequency is high enough (see Fig. 7). Since the phenotypic state $s$ [definition (6)] changes both the mode $m_s$ and the breadth $b_s$ of the tolerance curve, we introduce the following shorthand notation for the phenotype $(m_s,b_s)$,

$$\begin{aligned} mb(s)= m_s,b_s. \end{aligned}$$

(8)

1.2 A.2 Calculating the fitness in a simplified scenario

The stress intensities of all stress events are first assumed to be constant $\psi$, that is $\sigma _\phi =0$. We define environmental states $\varPhi (\gamma )$ by environmental settings

$$\begin{aligned} \gamma \in \{ 0,1 \}, \end{aligned}$$

(9)

where stress is indicated by $\gamma =1$ and the stress-free environment by $\gamma =0$,

$$\begin{aligned} \varPhi (\gamma )= {\left\{ \begin{array}{ll} \psi &{}\quad \text{ if } \gamma =1 \\ 0 &{}\quad \text{ if } \gamma =0. \end{array}\right. } \end{aligned}$$

(10)

In order to avoid repeating similar calculations, we consider a sequence of phases with alternating environmental settings and lengths $t_1$ and $t_2$ (see Fig. 8). It is arbitrary whether the phases of length $t_1$ have environmental setting $\gamma =1$ and the phases of length $t_2$ have environmental setting $\gamma =0$ or vice versa. The environmental setting of the phases with length $t_1$ is denoted by $\gamma _0$ and the environmental setting of the phases with length $t_2$ is $1-\gamma _0$. The length of a life was set to be $1$ and for the time being we additionally assume that the length of the first phase is not longer than lifetime, $t_1 \le 1$. We assume that organisms are born at uniformly distributed points during the first phase.

Fig. 8

The environmental course shown in Fig. 7 is abstracted into a sequence of phases with alternating environmental setting ($\gamma _0$ and $1-\gamma _0$) and length ($t_1$ and $t_2$). In this example, the interval $[t_1,1[$ is divided into $k=5$ intervals $A_j=[q_j,q_{j+1}[$ with alternating environmental setting [cf. definition (13)]

With this simplification, the lives of all organisms lie within the interval $[0,1+t_1[$. According to formula (3) in the main text the fitness of an individual is calculated by dividing the interval $[0,1+t_1[$ into subintervals with constant phenotypic state and environmental setting, and multiplying the performance during each subinterval, raised to the power of the subintervals length (since life length is $1$). Naturally, subintervals during which the individual is not alive do not contribute to the fitness. Since the fitness of an individual is calculated by the geometric mean of the contribution of the different phases of life, we can formally assume that the fitness during subintervals that are not experienced is $1$. According to formula (5), the fitness of the genotype (in short: the genotype fitness) is the geometric mean of the fitness of $n \rightarrow \infty$ individuals.

This is equivalent to first dividing the interval $[0,1+t_1[$ into a partition $\mathbb {A}$; then for each subinterval $A \in \mathbb {A}$, calculating the performance of a given individual during the subinterval; then calculating the geometric mean of such performances of $n \rightarrow \infty$ individuals (we term that the interval’s contribution to the genotype fitness $\tilde{w}(A)$); and finally assembling the genotype fitness as

$$\begin{aligned} \tilde{w}([0,1+t_1[)=\prod _{A \in \mathbb {A}} \tilde{w}(A). \end{aligned}$$

(11)

We divide the interval $[0,1+t_1[$ into three subintervals:

• $[0,t_1[$: Individuals are born at uniformly distributed points during that interval.

• $[t_1,1[$: All individuals experience that interval completely.

• $[1,1+t_1[$: Individuals die at uniformly distributed points during that interval.

These intervals are again divided such that each change of the environmental setting marks a division point. Once more we divide these smaller intervals into intervals during which the phenotype changes and intervals during which the phenotype stays constant. The genotype fitness contribution of these smallest intervals are the building blocks of which we will assemble the genotype fitness.

1.2.1 A.2.1 Interval $[0,t_1[$

The time of birth $u_i$ of an individual is uniformly distributed on $[0,t_1[$ and independent of the time of birth of other individuals. The environmental setting at birth, and consequently the phenotypic state during this first interval, is $\gamma _0$. Therefore the contribution of the interval $[0,t_1[$ to the fitness of that individual is $f(mb(\gamma _0),\varPhi (\gamma _0))^{t_1-u_i}$, and the contribution of the interval $[0,t_1[$ to the genotype fitness is

$$\begin{aligned} \tilde{w}([0,t_1[)&=\lim _{n \rightarrow \infty } \prod _{i=1}^n \left( f(mb(\gamma _0),\varPhi (\gamma _0))^{t_1-u_i}\right) ^\frac{1}{n}\nonumber \\&=f(mb(\gamma _0),\varPhi (\gamma _0))^{\lim _{n \rightarrow \infty }\sum _{i=1}^n \frac{1}{n} (t_1-u_i) }\nonumber \\&\mathop {=}\limits ^{(\star )}f(mb(\gamma _0),\varPhi (\gamma _0))^{E[t_1-u_i]}\nonumber \\&=f(mb(\gamma _0),\varPhi (\gamma _0))^{\frac{1}{2}t_1}, \end{aligned}$$

(12)

where $(\star )$ holds by the law of large numbers.

1.2.2 A.2.2 Interval $[t_1,1[$

Independently of the time of birth, all individuals experience the interval $[t_1,1[$ completely and with the same course of the phenotypic state (consider one of the two solid courses in Fig. 7). Therefore, the genotype fitness contribution of that interval equals the contribution of the interval to the fitness of one individual. To obtain that contribution, we divide the interval $[t_1,1[$ into subintervals

$$\begin{aligned} A_j=[q_j,q_{j+1}[ \end{aligned}$$

(13)

with altering environmental settings. The number $k$ of such subintervals can be obtained by adding the number of intervals with environmental setting $1-\gamma _0$ that start during $[t_1,1[$ to the number of intervals with environmental setting $\gamma _0$ that start during $[t_1+t_2,1[$,

$$\begin{aligned} k=\min \{n \in \mathbb {N} : n(t_1+t_2) \ge 1-t_1 \}+ \min \{n \in \mathbb {N} : n(t_1+t_2) \ge 1-t_1-t_2 \} \end{aligned}$$

(14)

The left boundary of the first interval $A_1$ is $q_1=t_1$ and the right boundary of the last interval $A_k$ is $q_{k+1}=1$. The other boundaries $q_j$ (for $j=2,\ldots ,k$) can be calculated recursively,

$$\begin{aligned} q_1= & {} t_1,\nonumber \\ \text {for } j=2,\ldots ,k:\ q_j= & {} q_{j-1}+{\left\{ \begin{array}{ll} t_2 &{}\quad \text{ if } j \text{ even } \\ t_1 &{}\quad \text{ if } j \text{ odd }, \end{array}\right. }\nonumber \\ q_{k+1}= & {} 1. \end{aligned}$$

(15)

The environmental setting during the phase $A_j$ is,

$$\begin{aligned} \gamma (A_j)={\left\{ \begin{array}{ll} \gamma _0 &{}\quad \text{ if } j \text{ even } \\ 1-\gamma _0 &{}\quad \text{ if } j \text{ odd }. \end{array}\right. } \end{aligned}$$

(16)

We use the shorthand notations $\gamma _j=\gamma (A_j)$ and $\phi _j=\varPhi (\gamma _j)$.

During the environmental setting $\gamma$, let $\vartheta (\gamma )$ be the rate with which the phenotypic state changes until the target state $\gamma$ is reached. This rate is equivalent to the signed speed with which the peak of the tolerance curve shifts, measured relative to the Euclidean distance between the peaks of the tolerance curves of the induced and the non-induced phenotype,

$$\begin{aligned} \vartheta (\gamma )=\left( (m_1-m_0)^2+\left( \frac{1}{\sqrt{2\pi } b_1}-\frac{1}{\sqrt{2\pi } b_0}\right) ^2\right) ^{-\frac{1}{2}} \cdot \left\{ \begin{array}{ll} v_1 &{}\quad \quad \text{ if } \gamma =1 \\ -v_0 &{}\quad \quad \text{ if } \gamma =0. \end{array} \right. \end{aligned}$$

(17)

We use the shorthand notation $\vartheta _j=\vartheta (\gamma _j)$.

The phenotypic state $s(q_1)$ at time $q_1$ is identical to the phenotypic state $\gamma _0$ at birth. The phenotypic states $s(q_j)$ at the other boundary points can be calculated recursively by adding the product of the adaptation rate $\vartheta _{j-1}$ and the length $|A_{j-1}|$ of the preceding interval to the phenotypic state $s(q_{j-1})$ at the previous boundary point, and constraining to the interval $[0,1]$,

$$\begin{aligned} s(q_1)= & {} \gamma _0\nonumber \\ \text {for } j \ne 1: s(q_j)= & {} \max \{0,\min \{1,s(q_{j-1})+\vartheta _{j-1}|A_{j-1}|\}\}. \end{aligned}$$

(18)

We use the shorthand notation $s_j=s(q_j)$.

With the phenotypic states $s_j$ and $s_{j+1}$ we can calculate the time $\delta (A_j)$ of transformation during the interval $A_j$,

$$\begin{aligned} \delta (A_j)=\min \left\{ | A_j | , \frac{s_{j+1}-s_j}{\vartheta _j} \right\} . \end{aligned}$$

(19)

We use the shorthand notation $\delta _j=\delta (A_j)$.

The phases with constant phenotype are infinitesimally short during a transformation. We divide the transformation interval $[q_j,q_j+\delta _j[$ into $n \rightarrow \infty$ equal steps. One step has length $\frac{\delta _j}{n}$ and the phenotypic state during the $q$’th step is $\frac{n-q}{n}s_j+\frac{q}{n}s_{j+1}$. Hence the genotype fitness contribution of the interval is

$$\begin{aligned} \tilde{w}([q_j,q_j+\delta _j[)= & {} \lim _{n\rightarrow \infty }\prod _{r=1}^n f\left( {mb}\left( {\frac{n-r}{n}s_j+\frac{r}{n}s_{j+1}}\right) ,{\phi _j}\right) ^{\frac{\delta _j}{n}}\nonumber \\= & {} g(mb(s_j),mb(s_{j+1}),\phi _j)^{\delta _j} \end{aligned}$$

(20)

The function $g$ is defined in a general form in formula (4) in Sect. 2 and further derived in Sect. A.4, whereby there additionally the area under the tolerance curve and the environmental state is changing linearly. For the application here those parameters are constant and hence their initial and terminal parameters are equal. Note again that the area $c$ under the curve is not included in the formulas here because it is normalized and the environmental state is represented by a single parameter since it stays constant during the considered phase.

During the remaining part $[q_j+\delta _j,q_{j+1}[$ of the interval $A_j$, the phenotypic state is constant $\gamma _j$ and hence the fitness contribution of the interval $[q_j+\delta _j,q_{j+1}[$ is

$$\begin{aligned} \tilde{w}([q_j+\delta _j,q_{j+1}[)=f(mb(\gamma _j), \phi _j)^{|A_j|-\delta _j}. \end{aligned}$$

(21)

Finally, the contribution of an interval $A_j$ can be assembled from its two parts,

$$\begin{aligned} \tilde{w}(A_j)=\tilde{w}([q_j,q_{j+1}[)=\tilde{w} ([q_j,q_j+\delta _j[)\tilde{w}([q_j+\delta _j,q_{j+1}[) \end{aligned}$$

(22)

and the contribution of the entire interval $[t_1,1[$ is

$$\begin{aligned} \tilde{w}([t_1,1[)=\prod _{j=1}^k \tilde{w}(A_j). \end{aligned}$$

(23)

1.2.3 A.2.3 Interval $[1,1+t_1[$

Since the lifetime of an organism is $1$ and the time of birth $u_i$ is uniformly distributed on $[0,t_1[$, an individual lives until the time $u_i+1$, which is uniformly distributed on $[1,1+t_1[$. Because both the interval $[1,1+t_1[$ and the phases with environmental setting $\gamma _0$ have length $t_1$, the interval $[1,1+t_1[$ can include at most three subintervals with alternating environmental settings,

$$\begin{aligned} A'_j=[q'_j,q'_{j+1}[ . \end{aligned}$$

(24)

The boundaries of the subintervals are:

$$\begin{aligned} q'_1= & {} q_{k+1}=1\nonumber \\ q'_2= & {} \min \{ 1+t_1 , q_{k-1} + t_1 + t_2 \}\nonumber \\ q'_3= & {} \min \{ 1+t_1 , q_k + t_1 + t_2 \}\nonumber \\ q'_4= & {} 1+t_1. \end{aligned}$$

(25)

For the case $k=0$ (that is when $t_1=1$) or $k=1$, we formally define $q_{-1}=-t_2$ and $q_0=0$. Note that $|A'_1|=0$ ($q'_1=q'_2$ ) if $\frac{1}{t_1+t_2} \in \mathbb {N}$ or $\frac{1-t_1}{t_1+t_2} \in \mathbb {N}_0$. The other intervals $A'_j$ can have length 0, too, depending on the values for $t_1$ and $t_2$.

The environmental setting during the interval $A'_1$ is the same as the environmental setting $\gamma _{k}$ at the end of $[t_1,1[$, and the environmental settings of the intervals $A'_j$ alter,

$$\begin{aligned} \gamma (A'_j)= {\left\{ \begin{array}{ll} \gamma _k &{}\quad \text{ if } j \text{ odd } \\ 1-\gamma _k &{}\quad \text{ if } j \text{ even }. \end{array}\right. } \end{aligned}$$

(26)

We use analogous shorthand notations as before, $\gamma '_j=\gamma (A'_j)$, $\vartheta '_j=\vartheta (\gamma '_j)$ and $\phi '_j=\varPhi (\gamma '_j)$.

Since $q'_1=q_{k+1}$, the phenotypic state $s(q'_1)$ is $s_{k+1}$ and the phenotypic states at the succeeding boundaries are calculated analogous to formula (18),

$$\begin{aligned} s(q'_1)= & {} s_{k+1}\nonumber \\ \text {for } j \ne 1: \ s(q'_j)= & {} \max \{0,\min \{1,s(q'_{j-1})+\vartheta '_{j-1}|A'_{j-1}|\}\}. \end{aligned}$$

(27)

We use the shorthand notation $s'_j=s(q'_j)$.

Analogous to formula (19), the phenotype is transforming during the interval $A'_j$ for a time

$$\begin{aligned} \delta (A'_j)=\min \left\{ | A'_j | , \frac{s'_{j+1}-s'_j}{\vartheta '_j} \right\} . \end{aligned}$$

(28)

We use the shorthand notation $\delta '_j=\delta (A'_j)$.

The probability that the point $u_i+1$ where a given life ends lies in the interval $[q'_j,q'_j+\delta '_j[$ during which the phenotype is transforming is

$$\begin{aligned} P(u_i+1 \in [q'_j,q'_j+\delta '_j[)=\frac{\delta '_j}{t_1}. \end{aligned}$$

(29)

Given that an individual’s life ends during the transformation $[q'_j,q'_j+\delta '_j[$, the exact time when the life ends is uniformly distributed on this interval. We divide the transformation into $n \rightarrow \infty$ steps. The step length is $\frac{\delta '_j}{n}$ and the contribution of the interval $[q'_j,q'_j+\delta '_j[$ to the fitness of an individual whose life ends at the $i$’th step is

$$\begin{aligned} \lim _{n \rightarrow \infty } \prod _{r=1}^i f\left( mb \left( \frac{n-r}{n}s'_j+\frac{r}{n}s'_{j+1} \right) ,\phi '_j \right) ^{\frac{\delta '_j}{n}}. \end{aligned}$$

(30)

We consider $n \rightarrow \infty$ individuals and due to the law of large numbers we can assume that during each step the life of one individual ends. Therefore the interval’s genotype fitness contribution, under the condition that life ends during this interval, is

$$\begin{aligned}&\tilde{w}([q'_j,q'_j+\delta '_j[ : u_i+1 \in [q'_j,q'_j+\delta '_j[)\nonumber \\&\quad =\lim _{n \rightarrow \infty } \prod _{i=1}^n \left( \prod _{r=1}^i f\left( mb \left( \frac{n-r}{n}s'_j+\frac{r}{n}s'_{j+1} \right) ,\phi '_j \right) ^{\frac{\delta '_j}{n}}\right) ^\frac{1}{n}\nonumber \\&\quad =h(mb(s'_j),mb(s'_{i+j}),\phi '_j)^{\delta '_j} \end{aligned}$$

(31)

The function $h$ is derived in a generalized form in Sect. A.5. As for the general formula for $g$, there additionally the area under the tolerance curve and the environmental state is changing linearly. Again for the application here those parameters are constant and hence their initial and terminal parameters are equal. As before the area $c$ under the curve is not included in the formulas because it is normalized and the environmental state is represented by a single parameter since it stays constant during the considered phase.

The probability that a given individual is alive after the interval $[q'_j,q'_j+\delta '_j[$ is

$$\begin{aligned} P(u_i+1 \ge q'_j+\delta '_j)=\frac{1+t_1-(q'_j+\delta '_j)}{t_1}. \end{aligned}$$

(32)

Under the condition that all individuals are alive after the transformation $[q'_j,q'_j+\delta '_j[$, the interval’s genotype fitness contribution is according to formula (20),

$$\begin{aligned} \tilde{w}([q'_j,q'_j+\delta '_j[:u_i+1 \ge q'_j+\delta '_j)=g(mb(s'_j),mb(s'_{j+1}),\phi '_j)^{\delta '_j}. \end{aligned}$$

(33)

The unconditioned genotype fitness contribution of the interval $[q'_j,q'_j+\delta '_j[$ is then the product of the two conditioned contributions, each raised to the corresponding probability,

$$\begin{aligned} \tilde{w}([q'_j,q'_j+\delta '_j[)= & {} \tilde{w}([q'_j,q'_j+\delta '_j[:u_i+1 \in [q'_j,q'_j+\delta '_j[)^{P(u_i+1 \in [q'_j,q'_j+\delta '_j[)}\nonumber \\&\cdot \, \tilde{w}([q'_j,q'_j+\delta '_j[:u_i+1 \ge q'_j+\delta '_j)^{P(u_i+1 \ge q'_j+\delta '_j)}. \end{aligned}$$

(34)

During the remaining part $[q'_j+\delta '_j,q'_{j+1}[$ of the interval $A'_j$ the phenotypic state is constant $\gamma '_j$.

The probability that a given life ends during that part is

$$\begin{aligned} P(u_i+1 \in [q'_j+\delta '_j,q'_{j+1}[)=\frac{q'_{j+1}-(q'_j+\delta '_j)}{t_1}. \end{aligned}$$

(35)

Under the condition that all lives end during the interval $[q'_j+\delta '_j,q'_{j+1}[$, the individual lives end at points that are uniformly distributed on this interval. The interval’s genotype fitness contribution can hence be derived analogously to formula (12),

$$\begin{aligned} \tilde{w}([q'_j+\delta '_j,q'_{j+1}[:u_i+1\in [q'_j+ \delta '_j,q'_{j+1}[)=f(mb(\gamma '_j),\phi '_j)^{\frac{1}{2} (|A'_j|-\delta '_j)}. \end{aligned}$$

(36)

The probability that a given individual is alive after the interval $[q'_j+\delta '_j,q'_{j+1}[$ is

$$\begin{aligned} P(u_i+1 \ge q'_{j+1})=\frac{1+t_1-q'_{j+1}}{t_1}. \end{aligned}$$

(37)

Under the condition that individuals are alive after the interval $[q'_j+\delta '_j,q'_{j+1}[$, the interval’s genotype fitness is

$$\begin{aligned} \tilde{w}([q'_j+\delta '_j,q'_{j+1}[: u_i+1 \ge q'_{j+1} [)=f(mb(\gamma '_j),\phi '_j)^{|A'_j|-\delta '_j}. \end{aligned}$$

(38)

The unconditioned genotype fitness contribution of the interval $[q'_j+\delta '_j,q'_{j+1}[$ is therefore

$$\begin{aligned}&\tilde{w}([q'_j+\delta '_j,q'_{j+1}[)\nonumber \\&\quad =\tilde{w}([q'_j +\delta '_j,q'_{j+1}[ : u_i+1 \in [q'_j+\delta '_j,q'_{j+1}[)^{P(u_i+1 \in [q'_j+\delta '_j,q'_{j+1}[)} \nonumber \\&\qquad \cdot \, \tilde{w}([q'_j+\delta '_j,q'_{j+1} [ : u_i+1 \ge q'_{j+1})^{P(u_i+1 \ge q'_{j+1})}. \end{aligned}$$

(39)

Finally, the fitness contribution of an interval $A'_j$ can be assembled from its two parts,

$$\begin{aligned} \tilde{w}(A'_j)=\tilde{w}([q'_j,q'_j+\delta '_j[) \cdot \tilde{w}([q'_j+\delta '_j,q'_{j+1}[), \end{aligned}$$

(40)

and the genotype fitness contribution of the entire interval $[1,1+t_1[$ is

$$\begin{aligned} \tilde{w}([1,1+t_1[)=\prod _{j=1}^3 \tilde{w}(A'_j). \end{aligned}$$

(41)

1.2.4 A.2.4 The complete interval $[0,t_1+1[$

We now assemble the genotype fitness contributions of the subintervals and so obtain the genotype fitness. It is a function of the lengths $t_1$ and $t_2$ of the phases with alternating environmental setting and the environmental setting $\gamma _0$ of the first phase. We still consider the case that $t_1 \le 1$,

$$\begin{aligned} \tilde{w}_{t_1 \le 1}(t_1,t_2,\gamma _0)=\tilde{w}([0,1+t_1[)=\tilde{w}([0,t_1[) \cdot \tilde{w}([t_1,1[) \cdot \tilde{w}([1,1+t_1[). \end{aligned}$$

(42)

1.3 A.3 Generalization to the original scenario

We now relax the assumptions of Sect. A.2 and generalize the introduced method to calculate the genotype fitness for the original scenario where the first phase can last longer then a lifetime ($t_1>1$) and stress intensities are distributed randomly ($\sigma _\phi >0$), see Sect. 2.

1.3.1 A.3.1 Generalization for $t_1>1$

We excluded the case $t_1>1$ (see Fig. 9). In that case, organisms born during $[0,t_1-1[$ experience the environmental setting $\gamma _0$ during their whole life and consequently have the fitness $f(mb(\gamma _0),\varPhi (\gamma _0))$. Individuals that are born during $[t_1-1,t_1[$ experience the same environmental distribution as if the length of the first phase with constant environmental setting was $1$ instead of $t_1$. Under the condition that all individuals are born during that interval, the genotype fitness is therefore $\tilde{w}_{t_1 \le 1 }(1,t_2,\gamma _0)$. The general genotype fitness is then the product of the conditioned genotype fitness expressions, each raised to the power of the birth probability in the according interval, $\frac{t_1-1}{t_1}$ or $\frac{1}{t_1}$. For all $t_1>0$, the generalized formula for the genotype fitness is hence

$$\begin{aligned} \tilde{w}_{t_1 >0 } (t_1,t_2,\gamma _0)={\left\{ \begin{array}{ll} \tilde{w}_{t_1 \le 1}(t_1,t_2,\gamma _0) &{}\quad \text{ if } t_1 \le 1\\ f(mb(\gamma _0),\varPhi (\gamma _0))^\frac{t_1-1}{t_1} \tilde{w}_{t_1 \le 1}(1,t_2,\gamma _0)^\frac{1}{t_1} &{}\quad \text{ if } t_1>1. \end{array}\right. } \end{aligned}$$

(43)

Fig. 9

Sequence of phases with alternating environmental settings ($\gamma _0$ and $1-\gamma _0$) and lengths ($t_1$ and $t_2$). When $t_1>1$, individuals born during $[0,t_1-1[$ experience the environmental setting $\gamma _0$ during their whole life

1.3.2 A.3.2 Generalization for $\sigma _\phi >0$

We now relax the assumption that all stress events have the same stress intensity ($\sigma _\phi =0$) and instead assume a random distribution of stress intensities with known expectation value $\overline{\phi }$ and variance $\sigma _\phi ^2$. In the formulas, we can simply replace the stress intensity $\psi$ by $\overline{\phi }$ and $\psi ^2$ by $\sigma _\phi ^2+\overline{\phi }^2$ after expanding all terms that are connected to $\psi$. This can be verified by calculating the genotype fitness contribution of a stressful interval $A$ with constant phenotype $(m,b)$ as the geometric mean of the contribution of $A$ to the fitness of $n \rightarrow \infty$ individuals with independently drawn stress intensities $\psi _i$,

$$\begin{aligned} \tilde{w}(A)&=\lim _{n\rightarrow \infty }\prod _{i=1}^n f(m,b,\psi _i)^{\frac{1}{n} |A| } \nonumber \\&=\lim _{n\rightarrow \infty }\prod _{i=1}^n \left( \frac{1}{\sqrt{2 \pi }b}\mathrm {e}^{-\frac{1}{2}\frac{m^2-2m\psi _i+{\psi _i}^2}{b^2}}\right) ^{\frac{1}{n} |A| }\nonumber \\&=\frac{1}{\sqrt{2 \pi }b}\mathrm {e}^{-\frac{1}{2}\frac{\frac{1}{n}\sum _{i=1}^n\left( m^2-2m\psi _i+{\psi _i}^2\right) }{b^2} |A| }\nonumber \\&\mathop {=}\limits ^{(\star )}\frac{1}{\sqrt{2 \pi }b}\mathrm {e}^{-\frac{1}{2}\frac{m^2-2m\overline{\phi }+\sigma _\phi ^2+\overline{\phi }^2}{b^2} |A| }, \end{aligned}$$

(44)

where $(\star )$ holds by the law of large numbers.

Technically, the terms $\psi$ and $\psi ^2$ are substituted by $\overline{\phi }$ and $\sigma _\phi ^2+\overline{\phi }^2$, respectively, in formula (1) of the tolerance curve $f$; then the formulas for $g$ (20) and $h$ (31) need to be recalculated. Alternatively, the exchange of the terms can be realized directly in the formulas for the functions $f$, $g$ and $h$. In all other formulas the terms with $\psi$ only interact additively and hence they do not need to be changed directly. Note that stochasticity of the stress-free state could be introduced in the same way.

1.3.3 A.3.3 The original temporal distribution of stress

We can finally assemble the fitness of a genotype in the originally introduced environmental distribution, where stress events occur with frequency $p$ and persist for a time $t$ (hence the gaps between the stress events have length $\frac{1}{p}-t$), and where the individual times of birth are uniformly distributed within the sequence of stress events and stress-free times. Under the condition that all organisms are born during a stress event, the exact times of birth are uniformly distributed on the stress event. The genotype fitness under that condition is obtained by formula (43) with $t_1=t$, $t_2=\frac{1}{p}-t$ and $\gamma _0=1$. Given all organisms are born during the gap between two stress events, the exact times of birth are uniformly distributed on the gap. The genotype fitness under that condition is formula (43) with $t_1=\frac{1}{p}-t$, $t_2=t$ and $\gamma _0=0$. Altogether, the genotype fitness (here with all parameters as arguments) is then,

$$\begin{aligned}&w(p,t,\overline{\phi },\sigma _\phi ,v_0,v_1,m_0,m_1,b_0,b_1)\nonumber \\&\quad =\tilde{w}_{t_1>0}\left( t,\frac{1}{p}-t,1\right) ^{p t} \cdot \tilde{w}_{t_1 >0 }\left( \frac{1}{p}-t,t,0\right) ^{1-pt}. \end{aligned}$$

(45)

where $pt$ and $1-pt$ are the probabilities for a birth during stress or during a stress-free time, respectively.

1.4 A.4 Derivation of the function $g$

Here, we derive an expression for the function $g$, formula (4) in Sect. 2 in the main text. The formula is applied in the calculation (20) where the area under the tolerance curve is normalized, $c_a=c_z=1$, and the environmental state does not change during the transformation, $\phi _a=\phi _z=\phi$. In (20) we hence use the short-hand notation $g(m_a,b_a,m_z,b_z,\phi )$ or $g(mb(a),mb(z),\phi )$. The general formula is

$$\begin{aligned}&g\left( m_a,b_a,c_a,\phi _a,m_z,b_z,c_z,\phi _z\right) \nonumber \\&\quad =\lim _{n\rightarrow \infty }\prod _{r=1}^n f\left( \frac{n-r}{n}m_a+\frac{r}{n}m_z, \frac{1}{\frac{n-r}{n}\frac{1}{b_a}+\frac{r}{n}\frac{1}{b_z}}, \frac{n-r}{n}c_a+\frac{r}{n}c_z, \frac{n-r}{n}\phi _a+\frac{r}{n}\phi _z \right) ^\frac{1}{n}\nonumber \\&\quad =\lim _{n\rightarrow \infty }\prod _{r=1}^n\left( \frac{\left( \frac{n-r}{n}c_a+\frac{r}{n}c_z\right) \left( \frac{n-r}{n}\frac{1}{ b_a}+\frac{r}{n}\frac{1}{b_z}\right) }{\sqrt{2 \pi }} \mathrm {e}^{-\frac{1}{2}{\left( \frac{n-r}{n}m_a+\frac{r}{n}m_z-(\frac{n-r}{n}\phi _a+\frac{r}{n}\phi _z)\right) ^2\left( \frac{n-r}{n}\frac{1}{ b_a}+\frac{r}{n}\frac{1}{b_z}\right) ^2}}\right) ^\frac{1}{n}\nonumber \\&\quad =\frac{1}{\sqrt{2 \pi }}\lim _{n\rightarrow \infty }\prod _{r=1}^n \left( \left( \frac{n-r}{n}c_a+\frac{r}{n}c_z\right) ^\frac{1}{n} \left( \frac{n-r}{n}\frac{1}{ b_a}+\frac{r}{n}\frac{1}{b_z}\right) ^\frac{1}{n} \right) .\nonumber \\&\qquad \cdot \mathrm {e}^{-\frac{1}{2}\lim _{n\rightarrow \infty }\sum _{r=1}^n\frac{1}{n}{\left( \frac{n-r}{n}m_a+\frac{r}{n}m_z-(\frac{n-r}{n}\phi _a+\frac{r}{n}\phi _z)\right) ^2\left( \frac{n-r}{n}\frac{1}{ b_a}+\frac{r}{n}\frac{1}{b_z}\right) ^2}}\nonumber \\&\quad =\frac{1}{\sqrt{2 \pi }} \zeta _{g1} \zeta _{g2} \mathrm {e}^{-\frac{1}{2}\zeta _{g3}} \end{aligned}$$

(46)

where $\zeta _{g1}=c_a$ if $c_a=c_z$ and otherwise

$$\begin{aligned} \zeta _{g1}= & {} \lim _{n\rightarrow \infty } \prod _{r=1}^n\left( \frac{n-r}{n}c_a+\frac{r}{n}c_z\right) ^{\frac{1}{n}}\nonumber \\= & {} \exp \left( \lim _{n\rightarrow \infty } \log \left( \prod _{r=1}^n\left( \frac{n-r}{n}c_a+\frac{r}{n}c_z\right) ^{\frac{1}{n}}\right) \right) \nonumber \\= & {} \exp \left( \lim _{n\rightarrow \infty }\sum _{r=1}^n \frac{1}{n} \log \left( \frac{n-r}{n}c_a+\frac{r}{n}c_z\right) \right) \nonumber \\= & {} \exp \left( \frac{1}{c_z-c_a} \int _{c_a}^{c_z} \log (x) \, \mathrm {d}x\right) \nonumber \\= & {} \exp \left( \frac{1}{c_z-c_a} \left( c_z \log (c_z)+c_a-c_a \log (c_a)-c_z \right) \right) . \end{aligned}$$

(47)

Accordingly $\zeta _{g2}=\frac{1}{b_a}$ if $b_a=b_z$ and otherwise

$$\begin{aligned} \zeta _{g2}= & {} \lim _{n\rightarrow \infty } \prod _{r=1}^n\left( \frac{n-r}{n}\frac{1}{b_a}+\frac{r}{n}\frac{1}{b_z}\right) ^{\frac{1}{n}}\nonumber \\= & {} \exp \left( \frac{1}{\frac{1}{b_z}-\frac{1}{b_a}} \left( \frac{1}{b_a} \log (b_a)+\frac{1}{b_a}-\frac{1}{b_z} \log (b_z)-\frac{1}{b_z} \right) \right) , \end{aligned}$$

(48)

and

$$\begin{aligned} \zeta _{g3}= & {} \lim _{n\rightarrow \infty }\sum _{r=1}^n{\frac{1}{n} \left( \frac{n-r}{n}m_a+\frac{r}{n}m_z-\left( \frac{n-r}{n}\phi _a+ \frac{r}{n}\phi _z\right) \right) ^2\left( \frac{n-r}{n} \frac{1}{ b_a}+\frac{r}{n}\frac{1}{b_z}\right) ^2}\nonumber \\= & {} \int _0^1\left( (1-x)m_a+x m_z-((1-x)\phi _a+x \phi _z)\right) ^2\left( \frac{1-x}{ b_a}+\frac{x}{b_z}\right) ^2 \, \mathrm {d}x\nonumber \\= & {} \frac{1}{30 b_a^2 b_z^2}\left[ b_z^2 (6 m_a^2+m_z^2-3 m_z \phi _a +6 \phi _a^2+3 m_a (m_z-4 \phi _a-\phi _z)\right. \nonumber \\&-2 m_z \phi _z+3 \phi _a \phi _z+\phi _z^2)+b_a b_z (3 m_a^2+3 m_z^2-4 m_z \phi _a +3 \phi _a^2\nonumber \\&+\,m_a (4 m_z-6 \phi _a-4 \phi _z)-6 m_z \phi _z+4 \phi _a \phi _z+3 \phi _z^2)\nonumber \\&+\,b_a^2 (m_a^2+6 m_z^2+\phi _a^2+m_a (3 m_z-2 \phi _a-3 \phi _z)\nonumber \\&\left. +\,3 \phi _a \phi _z+6 \phi _z^2-3 m_z (\phi _a+4 \phi _z))\right] . \end{aligned}$$

(49)

1.5 A.5 Derivation of the function $h$

Here, we derive an expression for the function $h$ which is introduced in the calculation (20). In the generalized form here, all parameters of the tolerance curve, $m$, $\frac{1}{b}$ and $c$, and the environmental state $\phi$ change linearly from $m_a$, $\frac{1}{b_a}$, $c_a$, $\phi _a$ to $m_z$, $\frac{1}{b_z}$, $c_z$, $\phi _z$. In the calculation (20) the area under the tolerance curve is normalized, $c_a=c_z=1$, and the environmental state is not changing during the transformation, $\phi _a=\phi _z=\phi$, and we hence use the short-hand notation $h(m_a,b_a,m_z,b_z,\phi )$ or $h(mb(a),mb(z),\phi )$. The general formula is given by

$$\begin{aligned}&h \left( m_a,b_a,c_a,\phi _a,m_z,b_z,c_z,\phi _z\right) \nonumber \\&\quad =\lim _{n \rightarrow \infty } \prod _{i=1}^n \left( \prod _{r=1}^i f \left( \frac{n-r}{n}m_a+\frac{r}{n}m_z, \frac{1}{\frac{n-r}{n}\frac{1}{b_a}+\frac{r}{n}\frac{1}{b_z}}, \frac{n-r}{n}c_a+\frac{r}{n}c_z, \frac{n-r}{n}\phi _a+\frac{r}{n}\phi _z \right) ^\frac{1}{n} \right) ^\frac{1}{n}\nonumber \\&\quad =\lim _{n \rightarrow \infty } \prod _{i=1}^n f\left( \frac{n-i}{n}m_a+\frac{i}{n}m_z, \frac{1}{\frac{n-i}{n}\frac{1}{b_a}+\frac{i}{n}\frac{1}{b_z}}, \frac{n-i}{n}c_a+\frac{i}{n}c_z, \frac{n-i}{n}\phi _a+\frac{i}{n}\phi _z \right) ^\frac{n-i}{n^2}\nonumber \\&\quad =\lim _{n \rightarrow \infty }\prod _{i=1}^n f\left( \frac{i}{n}m_a+\frac{n-i}{n}m_z, \frac{1}{\frac{i}{n}\frac{1}{b_a}+\frac{n-i}{n}\frac{1}{b_z}}, \frac{i}{n}c_a+\frac{n-i}{n}c_z, \frac{i}{n}\phi _a+\frac{n-i}{n}\phi _z \right) ^\frac{i}{n^2}\nonumber \\&\quad =\lim _{n\rightarrow \infty }\prod _{i=1}^n \left( \frac{ \left( \frac{i}{n}c_a+\frac{n-i}{n}c_z \right) \left( \frac{i}{n}\frac{1}{b_a}+\frac{n-i}{n}\frac{1}{ b_z} \right) }{\sqrt{2 \pi }} \mathrm {e}^{-\frac{1}{2}{\left( \frac{i}{n}m_a+\frac{n-i}{n}m_z-(\frac{i}{n}\phi _a+\frac{n-i}{n}\phi _z)\right) ^2\left( \frac{i}{n}\frac{1}{b_a}+\frac{n-i}{n}\frac{1}{b_z}\right) ^2}} \right) ^\frac{i}{n^2}\nonumber \\&\quad =\frac{1}{\root 4 \of {2 \pi }}\lim _{n\rightarrow \infty } \prod _{i=1}^n \left( \left( \frac{i}{n}c_a+\frac{n-i}{n}c_z \right) ^\frac{i}{n^2}\right) \left( \frac{i}{n}\frac{1}{b_a}+\frac{n-i}{n}\frac{1}{ b_z}\right) ^\frac{i}{n^2} \cdot \nonumber \\&\qquad \cdot \mathrm {e}^{-\frac{1}{2}\lim _{n\rightarrow \infty }\sum _{i=1}^n \frac{i}{n^2}\left( \frac{i}{n}m_a+\frac{n-i}{n}m_z- (\frac{i}{n}\phi _a+\frac{n-i}{n}\phi _z)\right) ^2 \left( \frac{i}{n}\frac{1}{b_a}+\frac{n-i}{n} \frac{1}{b_z}\right) ^2}\nonumber \\&\quad =\frac{1}{\root 4 \of {(2 \pi )}} \zeta _{h1} \zeta _{h2} \mathrm {e}^{-\frac{1}{2} \zeta _{h3}} \end{aligned}$$

(50)

where $\zeta _{h1}=\sqrt{c_a}$ if $c_a=c_z$ and otherwise

$$\begin{aligned} \zeta _{h1}= & {} \lim _{n\rightarrow \infty } \prod _{i=1}^n\left( \frac{i}{n}c_a+\frac{n-i}{n}c_z \right) ^{\frac{i}{n^2}}\nonumber \\= & {} \exp \left( \lim _{n\rightarrow \infty } \log \left( \prod _{i=1}^n\left( \frac{i}{n}c_a+\frac{n-i}{n}c_z\right) ^{\frac{i}{n^2}}\right) \right) \nonumber \\= & {} \exp \left( \lim _{n\rightarrow \infty }\sum _{i=1}^n \frac{1}{n}\frac{i}{n} \log \left( \frac{i}{n}c_a+\frac{n-i}{n}c_z\right) \right) \nonumber \\= & {} \exp \left( \frac{1}{c_a-c_z} \int _{c_z}^{c_a} \frac{x-c_z}{c_a-c_z} \log (x) \, \mathrm {d}x\right) \nonumber \\= & {} \exp \left( \frac{2 c_z^2 \log (c_z)-(c_a-3c_z)(c_a-c_z)+2 c_a (c_a - 2 c_z) \log (c_a)}{4 (c_a - c_z)^2} \right) . \end{aligned}$$

(51)

Accordingly $\zeta _{h2}=\sqrt{\frac{1}{b_a}}$ if $b_a=b_z$ and otherwise

$$\begin{aligned} \zeta _{h2}= & {} \lim _{n\rightarrow \infty } \prod _{i=1}^n\left( \frac{i}{n}\frac{1}{b_a}+\frac{n-i}{n}\frac{1}{b_z}\right) ^{\frac{i}{n^2}}\nonumber \\= & {} \exp \left( \frac{4 b_a b_z-3 b_a^2-b_z^2+2 b_z (2 b_a-b_z) \log (b_a)-2 b_a^2 \log (b_z)}{4 (b_a-b_z)^2}\right) , \end{aligned}$$

(52)

and

$$\begin{aligned} \zeta _{h3}= & {} \lim _{n\rightarrow \infty }\sum _{i=1}^n{\frac{i}{n^2}\left( \frac{i}{n}m_a+\frac{n-i}{n}m_z- \left( \frac{i}{n}\phi _a+\frac{n-i}{n}\phi _z \right) \right) ^2\left( \frac{i}{n}\frac{1}{ b_a}+\frac{n-i}{n} \frac{1}{b_z}\right) ^2}\nonumber \\= & {} \int _0^1 x \left( x m_a+(1-x) m_z- \left( \frac{i}{n}\phi _a+\frac{n-i}{n}\phi _z \right) \right) ^2\left( \frac{x}{ b_a}+\frac{1-x}{b_z}\right) ^2 \, \mathrm {d}x\nonumber \\= & {} \frac{1}{60 b_a^2 b_z^2} \left[ 2 b_a b_z (2 m_a^2 + m_z^2 + 2 \phi _a^2 + 2 m_a (m_z - 2 \phi _a - \phi _z) + 2 \phi _a \phi _z + \phi _z^2 - 2 m_z (\phi _a + \phi _z)) \right. \nonumber \\&+\, b_z^2 (10 m_a^2 + m_z^2 + 10 \phi _a^2 + 4 m_a (m_z - 5 \phi _a - \phi _z) + 4 \phi _a \phi _z + \phi _z^2 - 2 m_z (2 \phi _a + \phi _z)) \nonumber \\&+\, b_a^2 (m_a^2 + 2 m_z^2 + \phi _a^2 + 2 m_a (m_z - \phi _a - \phi _z) + \left. 2 \phi _a \phi _z + 2 \phi _z^2 - 2 m_z (\phi _a + 2 \phi _z))) \right] . \end{aligned}$$

(53)

1.6 A.6 Notations and abbreviations

This list contains a selection of notations and abbreviations. Notations which are only introduced and used in a short context are mostly not listed here.

Notation	Description	Introduction
		Section 2
$m$	Mode of the tolerance curve
$b$	Breadth of the tolerance curve
$c$	Area under the tolerance curve
$\phi$	Environmental state
$f$	Tolerance curve	(1)
$w_i$	Fitness of an individual	(3)
$g^\tau$	Fitness contribution of a phase with length $\tau$ during which the parameters change linearly	(4), (20)
$w$	Fitness of the genotype	(5), (45)
		Section 3
$m_0,b_0$	Mode and breadth of the non-induced tolerance curve
$m_1,b_1$	Mode and breadth of the induced tolerance curve
$s$	Phenotypic state
$v_0$	Adaptation speed of the peak of the tolerance curve in the direction of the peak of the non-induced tolerance curve
$v_1$	Adaptation speed of the peak of the tolerance curve in the direction of the peak of the induced tolerance curve
$p$	Frequency of stress occurrence
$t$	Duration of a stress event
$\overline{\phi }$	Expected stress intensity
$\sigma _\phi$	Standard deviation of the stress intensities
		Section 4
$d_0$	Instantaneous model: transforming delay from the induced to the non-induced phenotype
$d_1$	Instantaneous model: transforming delay from the non-induced to the induced phenotype
$v$	Continuous model: adaptation speed of the peak of the tolerance curve in either direction when $v_0=v_1$
$d$	Instantaneous model: transforming delay in either direction when $d_0=d_1$
$pt$	Proportion of time stress is present. Product of $p$ and $t$
		Appendix A
$mb(s)=m_s,b_s$	Mode and breadth of the tolerance curve at phenotypic state $s$	(8)
$\psi$	Stress intensity
$t_1,t_2$	Lengths of phases with constant environmental state
$\tilde{w}(A)$	Genotype fitness contribution of the interval $A$ of the time line in the simplified scenario, with $t_1 \le 1$	(11)
$\tilde{w}(A:E)$	Same as $\tilde{w}(A)$ but the individual times of birth are conditioned according to $E$
$\tilde{w}_{t_1 \le 1}(t_1,t_2,\gamma _0)$	Genotype fitness in the simplified scenario with $t_1 \le 1$	(42)
$\tilde{w}_{t_1 > 0}(t_1,t_2,\gamma _0)$	Genotype fitness in the simplified scenario with arbitrary $t_1 > 0$	(43)
$\gamma$	Environmental setting	(9)
$\varPhi (\gamma )$	Environmental state as a function of the environmental setting	(10)
$A_j$	$[q_j,q_{j+1}[$, $j$’th interval with alternating environmental setting in $[t_1,1[$	(13)
$k$	Number of intervals with alternating environmental setting in $[t_1,1[$	(14)
$q_j$	j’th boundary point in $[t_1,1[$ between intervals with alternating environmental setting, left boundary point of $A_j$	(15)
$\gamma (A_j)$	Environmental setting of the interval $A_j$	(16)
$\gamma _j$	Abbreviation for $\gamma (A_j)$
$\phi _j$	Abbreviation for $\varPhi (\gamma _j)$
$\vartheta (\gamma )$	Adaptation rate of the phenotypic state in the environmental setting $\gamma$	(17)
$\vartheta _j$	Abbreviation for $\vartheta (\gamma _j)$
$s(q_j)$	Phenotypic state at the point $q_j$, abbreviation: $s_j$	(18)
$\delta (A_j)$	Time of phenotypic change in the interval $A_j$, abbreviation: $\delta _j$	(19)
$A'_j$	$[q'_j,q'_{j+1}[$, $j$’th interval with alternating environmental setting in $[1,1+t_1[$. Attention: the $A'_j$ can have length zero!	(24)
$q'_j$	j’th boundary point in ${[}1,1+t_1{[}$ between intervals with alternating environmental setting, left boundary of $A'_j$	(25)
$\gamma (A'_j)$	Environmental setting of the interval $A'_j$	(26)
$\gamma '_j$	Abbreviation for $\gamma (A'_j)$
$\phi '_j$	Abbreviation for $\varPhi (\gamma '_j)$
$\vartheta '_j$	Abbreviation for $\vartheta (\gamma '_j)$
$s(q'_j)$	Phenotypic state at the point $q'_j$, abbreviation: $s'_j$	(27)
$\delta (A'_j)$	Time of phenotypic change in the interval $A'_j$, abbreviation: $\delta '_j$	(28)
$h^\tau$	Fitness contribution of a phase with length $\tau$ during which the parameters change linearly and which is experienced up to on its length uniformly distributed points	(31)