Evolutionary rescue can prevent rate-induced tipping

Abstract

The transformation of ecosystems proceeds at unprecedented rates. Recent studies suggest that high rates of environmental change can cause rate-induced tipping. In ecological models, the associated rate-induced critical transition manifests during transient dynamics in which populations drop to dangerously low densities. In this work, we study how indirect evolutionary rescue—due to the rapid evolution of a predator’s trait—can save a prey population from the rate-induced collapse. Therefore, we explicitly include the time-dependent dynamics of environmental change and evolutionary adaptation in an eco-evolutionary system. We then examine how fast the evolutionary adaptation needs to be to counteract the response to environmental degradation and express this relationship by means of a critical rate. Based on this critical rate, we conclude that indirect evolutionary rescue is more probable if the predator population possesses a high genetic variation and, simultaneously, the environmental change is slow. Hence, our results strongly emphasize that the maintenance of biodiversity requires a deceleration of the anthropogenic degradation of natural habitats.

Introduction

Today, the world’s ecosystems are faced with unprecedented rates of climate change, habitat fragmentation and destruction (Travis 2003; Teyssèdre and Robert 2014; Oliver et al. 2015; Trathan et al. 2015; Frishkoff et al. 2016; Lee et al. 2017; Selwood et al. 2015; Parmesan 2006). In many cases, affected species are unable to counter the fast conversion of their living conditions, e.g., due to barriers preventing dispersal to more suitable habitats or insufficient time to adapt via phenotypic plasticity or genetic variations (Williams et al. 2008; Urban et al. 2016). Consequently, corresponding populations can shrink dramatically or, in the worst case, become extinct (Johnson et al. 2017). Such drastic changes of population densities are called regime shifts (Folke et al. 2004; Scheffer et al. 2001; Hastings et al. 2018; Wernberg et al. 2015) or tipping events (Ashwin et al. 2012; Wieczorek et al. 2011; Vanselow et al. 2019; Siteur et al. 2016).

In ecology, regime shifts have often been associated with the transition of an ecosystem from one stable state to another alternative stable state (Dwomoh and Wimberly 2017; van de Leemput et al. 2016; Gårdmark et al. 2015). So far, based on the existence of alternative stable states, three types of regime shifts or tipping events have been formulated, whose course can best be illustrated using potential landscapes (Scheffer et al. 2008) (see Fig. 1A-C): (i) bifurcation-induced tipping (B-tipping) where slowly changing environmental conditions alter the potential landscape until a formerly stable state (left valley) vanishes and the system state (ball) proceeds to an alternative stable state (right valley) (Fig. 1A) (Folke et al. 2004; Ashwin et al. 2012; ii) noise-induced tipping (N-tipping) where the state of the system (ball) is pushed out of the basin of attraction of one stable state (left valley) by fluctuations (noise, Fig. 1B) (Ashwin et al. 2012; Horsthemke and Lefever 1984); and finally (iii) shock-induced tipping (S-tipping) where a single large perturbation (shock, extreme event) kicks the system out of its current basin of attraction (Fig. 1C) (Halekotte and Feudel 2020).

However, when the speed of environmental change is sufficiently fast, a fourth critical transition is possible. In fact, in reference to the critical rate of change which has to be surpassed for this transition to occur, it is called rate-induced tipping (R-tipping) (Wieczorek et al. 2011). The mechanism behind rate-induced tipping is not universal. It can appear in the presence of alternative stable states, for instance when, due to the fast environmental change, the system state crosses a moving basin boundary (Ashwin et al. 2012; Neijnens et al. 2021) or drifts through a catastrophic bifurcation (Chaparro-Pedraza 2021; Ritchie et al. 2021; Alkhayuon et al. 2019). However, and this is essential for this work, R-tipping does not necessarily require the presence of alternative stable states—in contrast to the other three tipping mechanisms. We will exclusively consider this latter case in which the tipping ’only’ manifests in a temporarily exceptional behavior—e.g., a large unexpected excursion in the transient dynamics (Hastings et al. 2018)—while its long-term behavior is always determined by a unique stable state. Such transient excursions can nevertheless embrace dangerously, unexpected states as, for instance, a temporarily increase in CO$_2$-concentrations in the atmosphere (Wieczorek et al. 2011) or a rapid population collapse (Vanselow et al. 2019; Siteur et al. 2016). Accordingly, R-tipping does indeed denote a critical transition.

In order to demonstrate how R-tipping can manifest in unexpected—possibly dramatic—transient dynamics, we exemplary study the potential landscape of a two-dimensional system which is subjected to a gradual but comparatively fast environmental change (Fig. 1D–G). The change is realized by a parameter p(t) which drifts from an upper to a lower bound ($p_{max}$ and $p_{min}$, respectively) at the constant rate $r_p$. Importantly, the system holds a unique stable state for all values within $[p_{min},\; p_{max}]$. At $t=0$, the system is in equilibrium. Hence, the ball resides in the minimum of the potential (Fig. 1D, F). As soon as p(t) starts to change, due to the ’reaction time’ of the system, the potential slides beneath the ball creating a gap between the current system state (ball) and the stable state (the moving minimum of the potential). The size of this gap naturally depends on the rate $r_p$. If $r_p$ is low (D), the actual system state (ball and cyan trajectory) remains close to the moving stable state (minimum of potential and grey dashed line in Fig. 1D and E, respectively). We call this behavior tracking. By contrast, if the rate $r_p$ is high (Fig. 1F, G), the system (magenta trajectory) reveals a temporarily large excursion from the moving stable state (minimum and grey dashed line). Hence, between the low and high rates $r_p$, the transient dynamics has switched from tracking to a temporarily large rate-induced deviation from the moving stable state.

Despite the fact that the system inevitably settles on the unique stable state, such a temporary deviation can be significant or even dangerous. If, for instance, the system state represents the density of a population, a transient excursion can put the system below a critical extinction threshold $x_e$ (grey dotted line). Such a threshold can be seen as a roughly estimated value below which it is no longer reasonable to ignore demographic stochasticity (Gomulkiewicz and Holt 1995) and thus a population whose density falls below $x_e$ is severely endangered of becoming extinct. We use an extinction threshold to distinguish whether some system dynamics has shown tracking or tipping behavior: Whenever the system state (aka a population density) drops below the extinction threshold for any time interval, we say that the system has tipped. Since the rate of environmental change has to exceed a critical rate to cause R-tipping, the relation between different time scales—in particular between the time scale of environmental change and the time scale of the intrinsic ecosystem dynamics (the ’reaction time’)—is a crucial determinant for this transition to occur (Wieczorek et al. 2011).

The relation between different time scales is also key to another ecological phenomenon, called evolutionary rescue, where populations are able to rescue themselves from extinction by rapid evolution (Gomulkiewicz and Holt 1995; Bell 2013; Gonzalez et al. 2013; Bell and Gonzalez 2009; Gomulkiewicz and Shaw 2013; Carlson et al. 2014). Naturally, whether populations are able to adapt sufficiently fast to rapid environmental change depends on their interactions with other species (Lavergne et al. 2010; Lawrence et al. 2012; Van der Putten et al. 2010; Schoener et al. 2001; Bastille-Rousseau et al. 2018) and some studies have in fact treated evolutionary rescue in a multi-species context (Henriques and Osmond 2020; De Mazancourt et al. 2008; Petkovic and Colegrave 2019; Northfield and Ives 2013; Norberg et al. 2012; Kovach-Orr and Fussmann 2013; Osmond and de Mazancourt 2013), e.g., in predator–prey systems (Osmond et al. 2017; Yamamichi and Miner 2015; Yamamichi et al. 2019; Cortez and Yamamichi 2019). In one of these works, Yamamichi and Miner (2015) demonstrate that rapid evolution of a prey species alone can rescue its non-evolving predator from extinction. They call this mechanism indirect evolutionary rescue. The reverse situation—adaptive predators that are able to support prey persistence—is examined by Osmond et al. (2017). However, neither of the two does explicitly model environmental change by means of time-dependent parameters. Hence, both miss an essential point, namely that evolutionary rescue is the outcome of a race between rapid evolution and environmental change, as suggested by (Gonzalez et al. 2013; Bell and Gonzalez 2009; Uecker et al. 2014; Alexander et al. 2014; Svensson and Connallon 2019; Ferriere and Legendre 2013; Carlson et al. 2014).

In our work, we model this race against extinction by taking into account the dynamics of environmental change and the dynamics of evolution. To this end, we expose a prey population to a sufficiently fast environmental change at which it experiences a rate-induced collapse. Then we test whether a predator population, that is enabled to adapt to the declining prey population, can prevent this rate-induced collapse. Hence, our ultimate aim is to bring together rate-induced-tipping and indirect evolutionary rescue. Both processes are determined by a critical rate, when considered separately. Combining both processes will lead to an expression which relates the critical rate of environmental change to the speed of evolution.

Furthermore, just like rate-induced tipping, (indirect) evolutionary rescue occurs on time scales which are comparable to the intrinsic dynamics of ecosystems. Hence, it can only be observed in the transients of eco-evolutionary systems. However, most theoretical studies analyzing eco-evolutionary systems solely focus on changes of the long-term behavior (Yoshida et al. 2003; Cortez and Patel 2017; Cortez and Ellner 2010; Yamamichi et al. 2011). Consequently, these studies neglect low population densities during the transient dynamics due to rate-induced tipping and therefore also miss out on recognizing counteracting processes such as evolutionary rescue (Vanselow et al. 2019; Hastings 2001; Hastings 2004; Hastings et al. 2018). Our study overcomes this problem by focusing on the part of a time series which is essential for studying rate-induced tipping and evolutionary rescue, the transient dynamics.

The scope of our work can be outlined as follows: First, we sketch the idea of how rapid evolution of a predator’s trait can prevent a rate-induced prey collapse. We then introduce the complete eco-evolutionary predator–prey model. Based on this model, we demonstrate the occurrence of R-tipping in a scenario without evolution in which the trait is fixed to its minimum value. Afterwards, we show that rapid evolution of the predator’s trait can prevent this rate-induced collapse. In the following section, we demonstrate that the probability of rate-induced tipping decreases with increasing genetic variation. Finally, we consider the sensitivity of our results depending on the initial state of the system. In the last section, we discuss our results.

Fig. 1

Illustrating B-tipping (A), N-tipping (B), S-tipping (C) using a sketch of a two-dimensional potential landscape. (D, E) Tracking if environmental change is slow ($r_P$ low): the state of the system (light blue trajectory) remains close to the moving stable state (minimum of the potential (D) and grey dashed line (E)). (F, G) R-tipping if environmental change is fast ($r_P$ high): the system (magenta trajectory) shows a deviation from the moving stable state (minimum of the potential (D) and grey dashed line (E)). If the state of the system represents the density of a population, rate-induced tipping can cause a temporarily collapse below the extinction threshold $x_e$ (grey dotted line)

How indirect evolutionary rescue can prevent rate-induced tipping

In this section, we study under which premises indirect evolutionary rescue is able to prevent the rate-induced collapse of the prey. Therefore, we study a three-dimensional eco-evolutionary predator–prey system including rapid evolution of the predator’s trait $\alpha$. We model the trait dynamics (3) in accordance with the quantitative genetics approach derived in Lande (1982) and Abrams et al. (1993). In this framework, the rate of change of the mean trait value $\frac{\text {d}\alpha }{\text {d}\alpha }$ is proportional to the additive genetic variance of the predator population $r_V A(\alpha )$ and the fitness gradient $\frac{\partial g(x,y,K(t),\alpha , \epsilon )}{\partial \alpha }$. The three-dimensional eco-evolutionary system is given by:

$$\begin{aligned} \epsilon \frac{\text {d}x}{\text {d}x}&= f(x,y,K(t),\alpha ,\epsilon ) \end{aligned}$$

(1)

$$\begin{aligned} \frac{\text {d}y}{\text {d}y}&= g(x,y,K(t),\alpha ,\epsilon ) \end{aligned}$$

(2)

$$\begin{aligned} \frac{\text {d}\alpha }{\text {d}\alpha }&= r_V A(\alpha ) \frac{\partial g( x,y,K(t),\alpha ,\epsilon )}{\partial \alpha }. \end{aligned}$$

(3)

In the following, this predator–prey system is exposed to a gradual environmental change which we model by the time-dependent parameter K(t):

$$\begin{aligned} K(t) = {\left\{ \begin{array}{ll} K_0 - r_K t &{} \text {if} \;\, K(t) > K_{min}\\ K_{min} &{} \text {else}. \end{array}\right. } \end{aligned}$$

(4)

According to (4), K(t) decreases at the constant rate $r_K$ until a prescribed minimum $K_{min}>0$ is reached (the environmental change is temporary, see also Fig. 3C). Furthermore, we assume that the system (1)–(3) possesses a unique stable equilibrium state $\varvec{e}(K,\epsilon ) = (e_x(K,\epsilon ), e_y(K,\epsilon ))$ for all fixed values $K(t) = K \in [K_0\; K_{min}]$ resulting from $f(x,y,K,\epsilon ) = g(x,y,K,\epsilon ) = 0$. Since we study the system for environmental conditions K(t) that change in time, $\varvec{e}(K(t),\epsilon )$ changes its position in phase space over time as well. For this reason, the equilibrium $\varvec{e}(K(t),\epsilon )$ is called a moving equilibrium (Wieczorek et al. 2011) (moving minimum in Fig. 1D and F). Such equilibria are also often referred to as quasi-static which stems from the assumption that they would be adopted if the ecosystem dynamics evolved much faster than the environmental conditions (Ashwin et al. 2012). Ultimately, as $K(t) = K_{min}$ is reached, the system inevitably approaches the sole stable long-term state $\varvec{e}(K_{min}, \epsilon )$. We choose $K_{min}$ in such a way that the equilibrium prey densities $e_x(K(t), \epsilon )$ and $e_x(K_{min},\epsilon )$ remain above a defined extinction threshold $x_e$ for all times (see fig. 3).

If the system (1)–(3) is able to stay close to the moving stable state during the transient, we say that it is tracking. For the eco-evolutionary system, this tracking means that the predator adjusts its trait $\alpha$ to the moving equilibrium value $e_\alpha (K(t),\epsilon )$ sufficiently fast to enable the coexistence of predator and prey $x,y \approx e_{x,y}(K,\epsilon )$. In the light of the gradual environmental degradation, we consider the case of perfect tracking ($x,y = e_{x,y}(K,\epsilon )$) as the optimal course of the eco-evolutionary system. Therefore, from now on, we refer to the moving stable equilibrium state $\varvec{e}(K(t),\epsilon )$ as the optimum state $\varvec{e}_{\text {opt}}(K(t),\epsilon )$.

Whether the eco-evolutionary system can track the moving optimum state during the environmental change depends crucially on how fast $\alpha$ evolves in relation to the speed of environmental change $r_K$. The trait $\alpha$ changes fast when the additive genetic variance $r_V A(\alpha )$ is high (Cortez 2016; Cortez 2018). Notice that the additive genetic variance is the product of phenotypic variation $A(\alpha )$ and the genotypic or genetic variation expressed by the rate $r_V$. However, in the following, we assume that the speed of evolution is mainly determined by genetic variation $r_V$ (Carlson et al. 2014).

In the following, we sketch how gradual environmental change of K(t) at different rates can reveal in dangerously low densities of the prey population and how this collapse can be prevented by rapid evolution of the predator’s trait. In the beginning, we assume that the eco-evolutionary system is situated in the optimum state $\varvec{e}_{\text {opt}}(K(t),\epsilon )$: $x_0 = e_{\text {opt},x}(K_0,\epsilon ), y_0 = e_{\text {opt},y}(K_0,\epsilon )$ and $\alpha _0 = e_{\text {opt},\alpha }(K_0,\epsilon )$.

If the predator can only respond slowly to changing conditions ($r_V$ is low in Fig. 2A and B), we can observe tracking if the rate of environmental change $r_K$ is low and R-tipping if $r_K$ is high. In the case of tracking (Fig. 2A), the U-shaped curve remains above the extinction threshold $x > x_e$ (cyan curve) while it temporarily drops below the extinction threshold $x < x_e$ between $t_e$ and $t_r$ when R-tipping occurs (magenta dashed line in Fig. 2B).

The rate-induced collapse of the prey can be explained by the emerging deviation of the mean trait value of the predator population $\alpha (t)$ from the optimum mean trait value $e_{\text {opt},\alpha }(K(t),\epsilon )$. This deviation, which is called lag-load $\tau _\alpha (K(t),\epsilon ) = |\alpha (t) - e_{\text {opt},\alpha }(K(t),\epsilon )|$ (Maynard Smith 1976; Kopp and Matuszewski 2014), quantifies the goodness of adaptation of the predator population: The smaller the lag-load $\tau _\alpha (K(t),\epsilon )$, the better suited is the predator population to the current environmental conditions K(t).

In case of tracking, the lag-load $\tau _{\alpha ,\text {track}}(K(t),\epsilon )$ is much smaller than in the case of R-tipping $\tau _{\alpha ,\text {tip}}(K(t),\epsilon )$ (compare the length of the doubled-headed arrows in Fig. 2C and D). Such a maladaption of the predator provides a possible explanation of the rate-induced collapse of the prey. For instance, when the maladaption manifests in a predator population that is much more aggressive than it would be in the optimum $\alpha (t) > e_{\text {opt},\alpha }(K(t),\epsilon )$, then the prey is confronted with a correspondingly high predation pressure. This can ultimately push the prey density below the extinction threshold $x_e$ (Fig. 2B). If, however, the predator is able to adapt sufficiently fast to the shrinking prey population ($r_V$ is high), the lag-load is reduced to a value $\tau _{\alpha ,\text {res}}<\tau _{\alpha ,\text {tip}}$ (green curve in Fig. 2D). In our example, this lowers the predation pressure on the prey which prevents its rate-induced collapse.

Fig. 2

(A) The eco-evolutionary system (1)–(3) tracks: the prey population x declines but remains above the extinction threshold $x>x_e$. (B) R-tipping: the prey density x drops below the extinction threshold $x<x_e$ between $t_e$ and $t_r$ due to fast environmental changes. (C, D) Slow/fast decline of the prey causes a small/large lag-load $\tau _{\alpha ,\text {track}}$/$\tau _{\alpha ,\text {tip}}$. (E) Distance between the minimum prey density and the extinction threshold $x_e$ reduces if predator evolution is fast. (F) The formerly collapsed prey population is rescued by rapid evolution of its predator (D): indirect evolutionary rescue occurs

The eco-evolutionary predator–prey model

In the following, we demonstrate the interplay between rate-induced tipping and rapid evolution in a multi-species context using a paradigmatic ecosystem model. For this purpose, we replace the general equations (1)–(3) by the classical predator–prey system developed by Rosenzweig and MacArthur (Rosenzweig and MacArthur 1963). In accordance with (Cortez and Ellner 2010; Yamamichi et al. 2019), we neglect units and treat the eco-evolutionary system as a purely mathematical model:

$$\begin{aligned} \frac{\text {d}x}{\text {d}x}&= rx\left( 1-\frac{x}{K(t)}\right) - \frac{\alpha xy}{1+\beta x}\end{aligned}$$

(5)

$$\begin{aligned} \frac{\text {d}y}{\text {d}y}&= \gamma \frac{\alpha x y}{1+\beta x} - m(\alpha )y\end{aligned}$$

(6)

$$\begin{aligned} \frac{\text {d}\alpha }{\text {d}\alpha }&= r_V(\alpha -\alpha _{min})(\alpha _{max}-\alpha )\left( \frac{\gamma x}{1+\beta x} - 2\alpha \right) . \end{aligned}$$

(7)

Here, the prey x grows logistically with maximum per-capita growth rate r and time-dependent carrying capacity K(t). Predator growth follows a Holling-Type II functional response with maximum predation rate $\alpha /\beta$, half-saturation constant $1/ \beta$ and conversion efficiency $\gamma$. The predator’s mortality is assumed to be proportional to the mortality rate $m(\alpha )$ and the density of the predator y. The time scale separation between prey and predator $\epsilon$ is determined by the ratio between the predator’s mortality rate $m(\alpha )$ and the prey’s reproduction rate r: $\epsilon =\frac{m(\alpha )}{r}$. Because $\epsilon$ describes a ratio and is assumed to be $0< \epsilon < 1$, the prey population always reproduces $\epsilon ^{-1}$ times faster than its predator whose time scale is then given by 1.0.

In contrast to most other eco-evolutionary models, we explicitly model the temporal change of the environment by means of a continuously declining carrying capacity K(t) (Eq. (4)). The carrying capacity K is a suitable quantity as it defines the maximum prey density that the environment can ’carry’. Accordingly, K is strongly linked to environmental conditions such as the availability of food sources, breeding sites or the suitability of climatic conditions. As discussed in the previous section, K is diminished at a rate $r_K$ until a prescribed minimum $K_{min}$ is reached. This minimum ensures that the prey does not go extinct due to the optimum falling below the extinction threshold. The realized scenario is comparable to a time-dependent degradation of the habitat due to climate variations, land use change or landscape fragmentation where the prey population loses access to essential resources.

In the eco-evolutionary model, the dynamical (evolutionary) trait of the predator is represented by its attack rate $\alpha$. We assume that the attack rate $\alpha$ affects both predation $\gamma \frac{\alpha xy}{1+\beta x}$ and mortality $m(\alpha ) = (\alpha ^2+c)y$ of the predator. Therefore, an adaptation of $\alpha$ is associated with a trade-off: If the predator invests in its attack rate $\alpha$, the predatory success will increase at the expense of a higher death rate (Abrams 2000; Cortez and Ellner 2010). In fact, it is this trade-off which enables a constructive adaptation of the predators attack rate depending on the available prey. Note that the trade-off crucially depends on how the attack rate $\alpha$ is incorporated into the mortality rate of the predator (see Appendix 12 for more details).

The dynamical equation for the attack rate $\alpha$ is derived according to Eq. (3). It is important to note that $\alpha$ depicts the mean trait within the predator population. Accordingly, the formulation of Eq. (7) is based on the assumption that the evolution of the mean trait is not significantly affected by the specific form of the frequency distribution of trait values within the predator population. By setting the phenotypic variance to $A(\alpha ) = (\alpha -\alpha _{min})(\alpha _{max}-\alpha )$, we restrict the adaptation of $\alpha$ to an interval bounded by the lowest $\alpha _{min}$ and highest $\alpha _{max}$ reasonable value. The genetic variation $r_V$ determines how fast the predator population is able to respond to changes (Cortez and Weitz 2014; Cortez 2018). Since evolutionary dynamics faster than the ecological dynamics are exceptional in natural systems (Cortez 2018), we only consider evolutionary dynamics which are similar to or slower than the slowest time scale in the ecosystem—here predator reproduction. Therefore, we restrict our analysis to $0<r_V \le 1.0$.

Overall, the four-dimensional eco-evolutionary system (5)–(7) contains four different time scales: the time scale of the prey’s life cycle, the time scale of the predator’s life cycle expressed by their relationship $\epsilon = \frac{m(\alpha )}{r} = \frac{\alpha ^2+c}{r}$, the rate of environmental change $r_K$ and the speed of evolution represented by the genetic variation $r_V$. The relationship between those time scales determines the dynamics of the system and will be the main focus of our study.

In the following, we discuss the dynamics of the eco-evolutionary system (5)–(7) using the exemplary parameter set given in Table 1. Notice, that the rate of environmental change $r_K$ and the speed of evolution $r_V$ will be varied throughout our study.

Table 1 Parameters and initial conditions of the eco-evolutionary system (5)–(7). Furthermore, important bifurcations and terms

Dynamics at the extreme trait values: no evolution but R-tipping

As has been discussed earlier, rate-induced tipping occurs as a system subjected to environmental change fails to track its moving stable state. In the context of eco-evolutionary dynamics, this moving stable state $\varvec{e}(K(t),\epsilon )$ represents the evolutionary optimum state $\varvec{e}_{\text {opt}}(K(t),\epsilon ) = \left( e_{x,\text {opt}}(K(t),\epsilon ), e_{y,\text {opt}}(K(t),\epsilon ) ,e_{\alpha ,\text {opt}}(K(t),\epsilon ))\right)$ for the given environmental condition K(t). Accordingly, in order to examine how R-tipping occurs in the eco-evolutionary system (5)–(7), we first need to determine its time-dependent optimum state $\varvec{e}_{\text {opt}}(K(t),\epsilon )$.

We start by determining the optimum attack rate $e_{\text {opt},\alpha }(K(t),\epsilon )$. Setting $\frac{\text {d}\alpha }{\text {d}\alpha }=0$ (7) reveals three different optimum attack rates $e_{\text {opt},\alpha }(K(t),\epsilon )$: two given by the extreme trait values (i) $\alpha =\alpha _{min}$ and (ii) $\alpha =\alpha _{max}$ and one by the attack rate $\alpha = \sqrt{c}$ for which the fitness gradient vanishes (iii) $\left( \frac{\gamma x}{1+\beta x} - 2\alpha \right) =0$. In this section, we concentrate on the two extreme cases which can only be maintained within the eco-evolutionary system if $\alpha (0) = \alpha _0 = \alpha _{min}$ or $\alpha (0)=\alpha _0 = \alpha _{max}$, respectively. Consequently, the trait $\alpha (t)$ remains constant for all time which ultimately excludes evolutionary adaptation. Though we are mainly interested in the case where evolution is active, it is quite instructive to demonstrate R-tipping without evolution first as it provides a benchmark for the following analyses.

As already stated, we only consider the dynamics for $0< \epsilon < 1.0$. However, for $\alpha =\alpha _{max}$, the time scale separation between predator and prey is given by $\epsilon = \frac{\alpha _{max}^2+c}{r} > 1$ and thus the predator would reproduce faster than its prey. Consequently, we neglect this case and focus only on the case $\alpha = e_{\alpha ,\text {opt}}(K(t),\epsilon ) = \alpha _{min}$. In this scenario, the moving optimum $\varvec{e}_{\text {opt}}(K(t),\epsilon )$ can be written as:

$$\begin{aligned} \varvec{e}_{\text {opt}}(K(t),\epsilon ) = {\left\{ \begin{array}{ll} \varvec{e}_{1,\text {opt}}(K(t),\epsilon ) &{} K_T < K(t) \ll K_H \\ \varvec{e}_{2,\text {opt}}(K(t),\epsilon ) &{} K_{min} \le K(t) \le K_T \end{array}\right. } \end{aligned}$$

(8)

with $\varvec{e}_{1,\text {opt}}(K(t),\epsilon ) = (n_x(\alpha _{min}), n_y(K(t), \alpha _{min}), K(t), \alpha _{min})$ and $\varvec{e}_{2,\text {opt}}(K(t),\epsilon ) = (K(t),0,K(t), \alpha _{min})$, where $n_x$ and $n_y$ fulfil ${\dot{x}}=0$ and ${\dot{y}}=0$ respectively (for $n_x$, $n_y$, $K_T$ and $K_H$ see Table 1). For large values of K(t), a Hopf bifurcation occurs at $K(t) = K_H$. However, in order to avoid long-term oscillations, we choose $K_0 \ll K_H$. The critical value $K_T$ marks the transcritical bifurcation at which $\varvec{e}_{1,\text {opt}}(K(t),\epsilon )$ loses stability, while $\varvec{e}_{2,\text {opt}}(K(t),\epsilon )$ becomes stable.

In Fig. 3, we illustrate how the optimum prey $\varvec{e}_{\text {opt},x}(K(t),\epsilon )$ (A) and predator densities $\varvec{e}_{\text {opt},y}(K(t),\epsilon )$ (B) change in time (gray dashed lines) due to their dependence on the time-varying parameter K(t). We find a constant optimum prey and a decreasing optimum predator population density as long as $K(t)>K_T$ (vertical dotted magenta line). Beyond the transcritical bifurcation $K_T$ the predator is extinct in the moving optimum state, while the optimum prey density decreases linearly in time as $\varvec{e}_{\text {opt},x}(K(t),\epsilon )=K(t)$ until $K_{min}$ is reached (horizontal magenta dotted line). Afterwards, the optimum prey density remains constant at the long-term state $\varvec{e}_{\text {opt},x}(K_{min},\epsilon )=K_{min}$. As K(t) decreases linearly in time, we expect the prey population to persist and the predator to go extinct if the eco-evolutionary system (5)–(7) is able to track the moving optimum state $\varvec{e}_{\text {opt}}(K(t),\epsilon )$ during the transient.

Fig. 3

How optimum prey $\mathbf{e} _{\text {opt},x}(K(t),\epsilon )$ (A) and predator density $\varvec{e}_{\text {opt},y}(K(t),\epsilon )$ (B) (gray dashed line) change in time. For $K(t)>K_T$, $e_{1,\text {opt}}(K(t),\epsilon )$ represents the moving optimum state. At $K_T$ (vertical magenta dotted line), $e_{1,\text {opt}}(K(t),\epsilon )$ and $e_{2,\text {opt}}(K(t),\epsilon )$ exchange stability in a transcritical bifurcation. For $K(t)<K_T$, $e_{2,\text {opt}}(K(t),\epsilon )$ is the moving optimum state. (C) Linear decline of the carrying capacity K(t) at the rate $r_K$ according to (4). Parameter: $x_0 = n_x(\alpha _{min})$, $y_0 = n_y(K(t),\alpha _{min})$, $K_0 = 10.0$, $\alpha _0 = \alpha _{min}$, $r_K = 0.05$, $r_V = 0$. Other parameters can be seen in Table 1

As depicted in Fig. 4, we now study the transient dynamics of the eco-evolutionary system (5)–(7) for the same initial condition $\varvec{e}_{\text {opt}}(K_0=10,\alpha _0=\alpha _{min})$ (8) but different rates of environmental change $r_K$. For a small rate of environmental change $r_K=0.05$ (A), we obtain the expected behavior, namely that the system is able to track the moving optimum (gray dashed line). As a consequence, the prey density x always remains above the extinction threshold $x_e$ while the predator population goes extinct in the long run. Interestingly, its extinction occurs much later than expected from the optimum predator density $\varvec{e}_{\text {opt},y}(K(t),\epsilon )$ (gray dashed line).

For a higher rate $r_K=1.0$ (Fig. 4B), we observe the rate-induced collapse of the prey. This collapse occurs because a small prey population, which constantly holds densities below its optimum $x<\varvec{e}_{\text {opt},x}(K(t),\epsilon )$, is confronted with a predator whose density is above its optimum $y>\varvec{e}_{\text {opt},y}(K(t),\epsilon )$. The resulting overwhelming predation pressure ultimately forces the prey population to drop below the extinction threshold $x_e$.

To clearly separate tracking $r_K<r_{crit}$ from R-tipping $r_K>r_{crit}$, we define the critical rate $r_{crit}$ as the lowest rate $r_K$ at which $x \le x_e$ during the transient dynamics. It should be noted that this definition is fundamentally different from the mathematical definition of R-tipping as outlined in (Ashwin et al. 2012; Wieczorek et al. 2011; Vanselow et al. 2019).

In the next section, we study whether we still find a rate-induced collapse of the prey if the predator population is capable of evolutionary adaptation. In such a case, the predator might be able to reduce the distance to the optimum state which would lower the predation pressure on the prey population and thereby possibly prevent the collapse.

Fig. 4

(A, B) The eco-evolutionary system (5)–(7) shows tracking (cyan) of the optimum state $\varvec{e}_{\text {opt}}(K(t),\epsilon )$ (gray dashed line) at low rate $r_K = 0.05$: prey density remains above extinction threshold $x_e$ (gray dotted line). (C, D) System shows R-tipping (magenta) at higher rate $r_K = 1.0$: prey density drops below extinction threshold $x_e$. (E) The critical rate $r=r_{crit}$ (black) is defined as the lowest rate $r_K$ at which $x \le x_e$ during the transient dynamics. The system tracks/tips for $r_K<r_{crit}$/$r_K>r_{crit}$. Parameter: $x_0 = n_x(\alpha _{min})$, $n_y(K_0,\alpha _{min})$, $K_0 = 10.0$, $\alpha _0 = \alpha _{min}$ and Table 1

Indirect evolutionary rescue prevents rate-induced tipping

In the following, we study whether a sufficiently fast evolutionary adaptation of the predator population can prevent the rate-induced collapse of the prey. Therefore, we choose an initial attack rate $\alpha _0$ between the two extreme values $\alpha _{min}$ and $\alpha _{max}$ and study the dynamics of the eco-evolutionary system (5)–(7) depending on the rate of environmental change $r_K$ and the genetic variation $r_V$. In this case, the eco-evolutionary system (5)–(7) possesses the moving optimum state $\varvec{e}_{\text {opt}}(K(t),\epsilon )$:

$$\begin{aligned} \varvec{e}_{\text {opt}}(K(t),\epsilon ) = {\left\{ \begin{array}{ll} \varvec{e}_{1,\text {opt}}(K(t),\epsilon ) &{} K_T < K(t) \ll K_H \\ \varvec{e}_{2,\text {opt}}(K(t),\epsilon ) &{} K_{min} \le K(t) \le K_T \end{array}\right. } \end{aligned}$$

(9)

with $\varvec{e}_{1,\text {opt}}(K(t),\epsilon ) = (n_x,n_y,K(t),\sqrt{c})$, $\varvec{e}_{2,\text {opt}}(K(t),\epsilon ) = \bigg (K(t),0,K(t),\frac{\gamma K(t)}{2+2\beta K(t)}\bigg )$ and $K_T = \frac{2c}{\gamma \sqrt{c}-2c\beta }$ respectively $K_H = \frac{4c}{\gamma \sqrt{c}-2c\beta }+\frac{1}{\beta }$ (see Table 1 for $n_x$, $n_y$ and other parameters). At $K(t)=K_T$, the system passes a transcritical bifurcation where $\varvec{e}_{1,\text {opt}}(K(t),\epsilon )$ and $\varvec{e}_{2,\text {opt}}(K(t),\epsilon )$ exchange stability (see Fig. 3).

When $r_V = 0.1$, as expected, increasing the rate of environmental change from $r_K=0.05$ (Fig. 5A) to $r_K=1.0$ (Fig. 5B) causes the rate-induced collapse of the prey. In the latter case the prey population is confronted with an extraordinary predation pressure because the attack rate very quickly becomes much higher than the optimum attack rate: $\alpha > \varvec{e}_{\text {opt},\alpha }(K(t),\epsilon )$ (Fig. 5K). As a consequence, the signed lag-load $\tilde{\tau }_\alpha (K(t),\epsilon ) = \alpha (t) - e_{\text {opt},\alpha }(K(t),\epsilon )$ is large and positive (Fig. 5K). Note that we show the signed lag-load $\tilde{\tau }_\alpha (K(t),\epsilon )$—instead of the usually applied absolute lag-load $\tau _\alpha$ (Gomulkiewicz and Holt 1995)—to emphasize the importance of the direction of maladaptation which is reflected by the sign of $\tilde{\tau }_\alpha (K(t),\epsilon )$. But as shown in figure 5C, this rate-induced collapse can be prevented if the genetic variation is high enough, here $r_V=0.9$ instead of $r_V=0.1$. The faster adaptation of the predator population suppresses the high initial attack rates. Instead, the attack rate $\alpha$ drops rather rapidly and attains rather low values—compared to the optimum and the scenarios representing tracking and R-tipping—before it approaches slowly the optimum. The mostly negative signed lag-load $\tilde{\tau }_\alpha (K(t),\epsilon )$ (Fig. 5) means that predators are less aggressive than in the optimum for most of the time which lowers the predation pressure on the prey and prevents its rate-induced collapse. As it is the predator whose adaptation secures the prey’s survival, it is a case of indirect evolutionary rescue.

In summary, we find three different transient dynamics: tracking, R-tipping and indirect evolutionary rescue depending on the relation of the rate of environmental change $r_K$ and the genetic variation $r_V$. In the next section, we evaluate this relationship in more detail.

Fig. 5

(A, B, C) Prey density x, (D, E, F) predator density y, (G, H, I) attack rate $\alpha$ of the eco-evolutionary system (5)–(7) and (J, K, L) signed lag-load $\tilde{\tau }_\alpha (K(t),\epsilon ) = \alpha (t) - e_{\text {opt},\alpha }(K(t),\epsilon )$ for different combination of the rate of environmental change $r_K$ and the rate of evolution $r_V$. The left column shows tracking (cyan), the middle column depicts R-tipping (magenta), and the right column demonstrates indirect evolutionary rescue (green). Parameters: $x_0 = e_{\text {opt},x}(K_0,\epsilon )$, $y_0 = e_{\text {opt},y}(K_0,\epsilon )$, $K_0=5.0$, $\alpha _0 = e_{\text {opt},\alpha }(K_0,\epsilon )$; other parameters can be seen in Table 1

The probability of rate-induced tipping decreases with increasing genetic variation

To obtain a comprehensive picture of rate-induced tipping and its prevention by indirect evolutionary rescue, we estimate the critical rate of environmental change $r_{crit}$ which has to be surpassed for R-tipping to occur depending on the rate of evolutionary adaptation based on genetic variation $r_V$. To start with, we evaluate whether solutions of the eco-evolutionary system (5)–(7) track ($x(t)>x_e$ for $t\rightarrow \infty$) or tip ($x(t)<x_e$ between $t_e$ and $t_r$) for rates of environmental change $r_K \in [0\; 1.5]$ and without rapid evolution $r_V = 0$. We find that the two possible outcomes are separated by a unique critical rate $r^0_{crit}$ (see left limit in Fig. 6). Following this, we evaluate for all rates $r_K \ge r^0_{crit}$ whether the system can change its behavior—from R-tipping to tracking—when the rate of evolution $r_V$ is increased ($r_V\in [0\;1]$). If this is the case, indirect evolutionary rescue occurs. Note that for $r_K<r^0_{crit}$, evolutionary rescue is not necessary to obtain tracking (cyan region Fig. 6).

As for $r_V=0$, for each $r_V$, a unique critical rate $r_{crit}$ can be determined which separates tipping (magenta) and tracking (green) behavior (Fig. 6). The corresponding curve (solid black line) portrays the expected relationship: the higher the genetic variation $r_V$, the lower the probability of R-tipping. Accordingly, a predator population with higher genetic variation is able to reduce its attack rate sufficiently fast to balance the gradual degradation of the prey’s carrying capacity (decrease of K(t)) and thus prevent its rate-induced collapse.

Fig. 6

Critical rate $r_{crit}$ (black curve) increases with rate of evolution $r_V$. It separates the $(r_K,r_V)$-parameter space into the R-tipping region (magenta) and the tracking-region (green/cyan). The black dotted line further divides the tracking-region into the area where evolutionary rescue is not necessary to obtain tracking (cyan) and the area where indirect evolutionary rescue enables tracking (green). Parameter $K_0 = 5.0$, $\alpha _0 = 0.6$, $x_0 = n_x(\alpha _0)$, $y_0 = n_y(\alpha _0,K_0)$. Other parameters can be seen in Table 1

The occurrence of indirect evolutionary rescue varies with initial conditions

Hereinafter, we examine if the appearance of R-tipping or indirect evolutionary rescue depends on the initial conditions of the eco-evolutionary system (5)–(7). So far, we have assumed that the system is always situated in the optimum state in the beginning. As a result, the predator population is optimally adapted before the carrying capacity starts to change. Consequently, the initial lag-load is given by $\tau _\alpha (K_0,\epsilon ) = 0$ as illustrated in figure 2. In the following, we vary the initial attack rate $\alpha _0$ and the initial value of the carrying capacity $K_0$ independently of one another. Hence, the predator population becomes less well adapted to the initial environmental condition $K_0$. To further ensure a reasonable relation between the initial prey and predator density $x_0$ and $y_0$, we set $x_0$ and $y_0$ to the optimum equilibrium densities $x_0=e_{1,\text {opt},x}(K_0,\epsilon )=n_x(\alpha _0)$ and $y_0=e_{1,\text {opt},y}(K_0,\epsilon )=n_y(K_0,\alpha _0)$ (see Table 1 for $n_x(\alpha )$ and $n_y(K(t),\alpha )$). Hence, the initial conditions ($n_x(\alpha _0),n_y(K_0,\alpha _0),K_0,\alpha _0$) are fully determined by setting the initial attack rate $\alpha _0$ and the initial carrying capacity $K_0$.

At first, we study exemplary initial conditions with different initial attack rates $\alpha _0 \gtrless e_{\text {opt},\alpha }(K_0,\epsilon )$ (Fig. 7). For $\alpha _0<e_{\text {opt},\alpha }(K_0,\epsilon )$, the higher the initial carrying capacity $K_0=4.5$ and the genetic variation $r_V=0.9$, the closer is the minimum of the prey density to the extinction threshold $x_e$ (Fig. 7A). For $\alpha _0>e_{\text {opt},\alpha }(K_0,\epsilon )$, the prey density shows the opposite behavior: instead of dropping, it increases in the beginning and reveals the highest distance to the extinction threshold for high values of $K_0$ and $r_V$, $K_0=4.5$ and $r_V=0.9$ (Fig. 7B). This contradiction can be explained by studying the initial densities of the predator population (Fig. 7C, D) and the change of the attack rate $\alpha$ (Fig. 7E, F). For $\alpha _0<e_{\text {opt},\alpha }(K_0,\epsilon )$, initial predator densities are consistently high while the attack rate increases. Consequently, the predation pressure increases leading to the early drop of the prey population density (Fig. 7G). Conversely for $\alpha _0>e_{\text {opt},\alpha }(K_0,\epsilon )$, initial densities of the predator population are much lower and the attack rate decreases. This reduces the predation pressure (Fig. 7H) causing the temporary increase of the prey density in the beginning. Hence, we expect prey populations with higher initial carrying capacity $K_0$ exposed to predators with $\alpha _0>e_{\text {opt},\alpha }(K_0,\epsilon )$ and higher genetic variation $r_V$ to be most likely to experience indirect evolutionary rescue.

Fig. 7

(A, B) Prey density x, (C, D) predator density y, (E, F) attack rate $\alpha$ and (G, H) predation pressure $pred = \frac{\alpha x y}{1+\beta x}$ for $\alpha _0<e_{\text {opt},\alpha }$ (left column) and $\alpha _0>e_{\text {opt},\alpha }$ (right column). Parameters: $\alpha _0 = e_{1,\text {opt},\alpha }\pm 0.1, x_0 = n_x(\alpha _0), y_0 = n_y(K_0,\alpha _0)$, $r_K = 0.1$. Other parameters Table 1

In Fig. 8, we now study the occurrence of evolutionary rescue for low (A, $r_V=0.2$) and high genetic variation (B, $r_V=0.9$) for initial conditions $K_0 \in [3,\; 5]$ and $\alpha _0 \in [0.1,\; 0.5]$. We choose these intervals to ensure the presence of the predator population and the absence of long-term oscillations. Therefore, initial conditions located in the gray regions are not taken into account.

In Fig. 8, the critical rate $r_{crit}$ of environmental change increases as the initial attack rate $\alpha _0$ achieves higher values. Accordingly, initial conditions with $\alpha _0>e_{\text {opt},\alpha }(K_0,\epsilon )$ are less prone to exhibit R-tipping. In this case, the prey population experiences lower predation pressure because (i) initial predator densities are lower and (ii) the attack rate decreases sufficiently fast (see Fig. 7F). As expected, these initial conditions experience indirect evolutionary rescue when the genetic variation is increased from $r_V = 0.2$ (A) to $r_V = 0.9$ (B). In the latter case, the already low predation pressure can be reduced even further due to a faster decreasing attack rate $\alpha$.

Interestingly, some initial conditions which are located in the tracking region for $r_V=0.2$ exhibit R-tipping as $r_V$ is increased to 0.9 (region I in Fig. 8B). This means that they tip precisely because of the evolutionary adaptation (increasing attack rate) being too fast which ultimately causes a higher predation pressure than for $r_V=0.2$ (compare Fig. 7G). A rather extreme case of this mechanism can be found in region II where the prey density has already dropped to densities below $x_e$ during the transient oscillation when $r_K=0$ (see Fig. 7 for the transient oscillation).

Furthermore, Fig. 8 demonstrates that the same initial lag-load $\tau _\alpha (K_0,\epsilon ) = |\alpha _0-e_{1,\text {opt},\alpha }(K_0,\epsilon )|$ can result in two completely different and contrasting outcomes: the occurrence of indirect evolutionary rescue or the rate-induced collapse of the prey population. This again emphasizes that aside from the magnitude of maladaptation, the direction of initial maladaptation—the signed lag-load $\tilde{\tau }_{\alpha }(K_0,\epsilon ) = \alpha _0-e_{1,\text {opt},\alpha }(K_0,\epsilon )$—can be a guiding quantity when studying the sensitivity of populations to the sudden onset of fast environmental changes.

Fig. 8

($K_0,\alpha _0$)-parameter space for $r_V = 0.2$ (A) and $r_V=0.9$ (B). Initial conditions in the colored region show R-tipping above the critical rate $r_{crit}$ (yellow: high, magenta: low). Initial conditions in the white non-colored region track. Initial conditions showing long-term oscillations or negative initial predator densities are not considered (gray regions). Initial conditions determined by higher initial attack rates experience indirect evolutionary rescue while initial conditions with lower initial attack rates change its behavior from tracking to R-tipping (colored region I). Initial conditions in II already achieve prey densities $x<x_e$ due to transient oscillations if $r_K=0$. Other parameters can be seen in Table 1

Discussion

The relation of environmental and evolutionary time scale

We have demonstrated that rapid evolution of a predator’s trait can prevent a rate-induced collapse of its prey. Whether this indirect evolutionary rescue is put into effect depends crucially on the relation between the rate of environmental change and the rate of evolutionary adaptation. We highlighted this relationship by deriving a critical rate of environmental change, above which R-tipping can be expected, depending on the genetic variation which determines the rate of evolution (see Fig. 6). We found that R-tipping is less and thus indirect evolutionary rescue is more likely when the genetic variation is high. This finding aligns well with former studies. For instance, Yamamichi and Miner (2015) who, however, did not explicitly consider rates of environmental change found that a higher genetic variation of a prey population shortens the vulnerable period of its non-evolving predator. Further theoretical (Gomulkiewicz and Holt 1995; Vander Wal et al. 2013; Barrett and Schluter 2008) and empirical studies (Ramsayer et al. 2013) show that high genetic variation increases the probability of evolutionary rescue of isolated populations.

As the rates of environmental change and evolutionary adaptation are of paramount importance for the outcome environmental changes, it is useful to consider under which circumstances they might be altered. For instance, populations might become more prone to R-tipping in the future as the speed of environmental change will increase due to climate change (Nadeau and Urban 2019; Shefferson et al. 2017; Jezkova and Wiens 2016). As suggested by the temporary change of the carrying capacity (Fig. 3C), even short-term changes of environmental conditions can potentially provide an adequate explanation for an observed population collapse. On the contrary, recent studies suggest that this trend could be counteracted by the dispersal of sub-populations between different patches (Bourne et al. 2014; Arumugam et al. 2020). As shown by Bourne et al. (2014), the underlying mechanism is that dispersal can increase the genetic variation of populations when beneficial mutations are spread across sub-populations. Furthermore, the recent study by Arumugam et al. (2020) demonstrates that even when evolutionary adaptation is neglected, the dispersal between predator–prey populations reduces the probability of R-tipping. However, this mechanism requires the existence of patches as well as a rather large overall population size. Furthermore, a loss of genetic variation occurs naturally if a population suddenly shrinks (Fountain et al. 2016; Baden et al. 2019). Even though, if this decline was transient, such populations would be more susceptible to rate-induced tipping in the following as its genetic variability would have already been degraded. Such maladaptations can ultimately lead to the extinction of a population but can also affect other populations which they interact with (i.e., their prey) (Yamamichi and Miner 2015; Osmond et al. 2017; Yamamichi et al. 2019).

Finally it should be noted that the specific relation between the rate of environmental change and the genetic variation is strongly linked to the choice of the extinction threshold. We introduced the extinction threshold in order to portray a population’s high endangerment when it undergoes stages of low density during its transient. One possibility to transfer this into the long-term behavior of the model would be the inclusion of the Allee effect in the dynamics of the prey (Allee et al. 1949). The Allee effect is useful in the context of eco-evolutionary systems because it incorporates a correlation between fitness and population size (Courchamp et al. 1999; Stephens et al. 1999; Stephens and Sutherland 1999).

Rapid evolution can stabilize or destabilize the transient dynamics

Furthermore, we have found that a sufficiently fast increase of the predator’s attack rate can cause indirect evolutionary rescue whereas its fast decline can lead to R-tipping (Fig. 7 and Fig. 8). Hence, rapid evolution of the predator’s trait can stabilize (by preventing low densities) or destabilize (by causing low densities) the transient state of the prey. Stabilizing and destabilizing evolution have so far been associated with the replacement of long-term oscillations by long-term stationary coexistence of populations and vice versa (Mougi and Iwasa 2010; Abrams 2000; Cortez and Patel 2017; Cortez et al. 2020). In this context, (indirect) evolutionary rescue and R-tipping can be seen as additional mechanisms which are able to stabilize or destabilize, respectively, the state of populations on ecologically relevant (short) time scales (during the transient). Interestingly, this stabilizing (destabilizing) effect of rapid evolution is independent of the long-term persistence of the predator population (compare Fig. 5 and Fig. 12). This again underlines the importance of considering transient states of populations instead of analyzing exclusively their long-term behavior, as has already been emphasized by Hastings et al. (2018).

The study of rate-induced tipping and evolutionary rescue in real ecosystems

So far, rate-induced tipping has not been observed or identified in nature. One reason is that determining whether a past tipping event was caused exclusively by too fast changing environmental conditions is quite challenging, especially since few empirical studies have a focus on the effect of rates of change in ecosystems (Chaparro-Pedraza 2021; Pinek et al. 2020). Due to this lack of data, the observation of its prevention by evolutionary rescue constitutes an even greater challenge. Additionally, most studies devoted to tipping in ecosystems rarely include information about trait changes of the organisms involved (Dakos et al. 2019). Hence, future empirical studies or experiments should consider rates of environmental change and, if possible, measure trait changes of the organisms involved. Moreover, there is a strong tendency in the literature to ascribe observed tipping to bifurcation-induced tipping based on the coexistence of alternative stable states, though a meta-analysis has revealed only weak evidence for that (Hillebrand et al. 2020).

In cases where empirical data is not available, simple mathematical models represent an appropriate tool for capturing the most relevant dynamics of ecosystems. Even if the eco-evolutionary system, as (5)–(7), is not based on real observations, it enables to analyze, at least qualitatively, the relationship between the rate of environmental change and the rate of evolution. Hence, the model which we have studied throughout this work can constitute the basis towards more realistic data-driven models which can be employed for examining the race between environmental change and evolution. So far, there only exist data-driven models or models with more realistic parameters that study R-tipping and which do not consider evolutionary processes on relevant time scales (Gil et al. 2020; van Nes et al. 2007; Neijnens et al. 2021). Hence, including the rate of evolution in these models represents a necessary next step for studying R-tipping in a more realistic framework.

Conclusion

All of our findings become solely visible in the transient dynamics of an eco-evolutionary model. Clearly, commonly applied models are an oversimplification of reality but as shown here, they can give valuable insights into the interaction of counteracting processes on ecologically relevant time scales. Our results show that we have to explicitly include rates of environmental change when modelling the race between evolutionary adaptation and extinction. Moreover, we suggest that besides the degree of maladaption (lag-load), its direction (signed lag-load)—trait lower/higher than optimum—is a decisive factor in inducing R-tipping or indirect evolutionary rescue. Finally, we once again emphasize the necessity of considering the transient dynamics because solely the transient reveals important processes on short time scales such as R-tipping or evolutionary rescue. Unraveling the mechanisms behind these processes will help us to understand the response of populations to habitat destruction or fragmentation and recent as well as future climate change.

Appendices

Time scale separation

To study the time scale separation between predator and prey, we rewrite the predator–prey system (10)–(11) in dimensionless prey density u, predator density v and dimensionless time $\hat{t}$:

$$\begin{aligned} \frac{\text {d}x}{\text {d}x}&= rx\left( 1-\frac{x}{K}\right) - \frac{\alpha xy}{1+\beta x}\end{aligned}$$

(10)

$$\begin{aligned} \frac{\text {d}y}{\text {d}y}&= \gamma \frac{\alpha x y}{1+\beta x} - m(\alpha )y. \end{aligned}$$

(11)

With $x = \frac{u m(\alpha )}{\alpha \gamma }$, $y=\frac{r}{\alpha } v$ and $t = r\hat{t}$, the predator–prey system can be written as:

$$\begin{aligned} \frac{\text {d}u}{\text {d}u}&= u\left( 1-\frac{u m(\alpha )}{\gamma \alpha K}\right) - \frac{uv}{1+\beta \frac{u m(\alpha )}{\alpha \gamma }}\end{aligned}$$

(12)

$$\begin{aligned} \frac{\text {d}v}{\text {d}v}&= \frac{m(\alpha )}{r} \left( \frac{uv}{1+\beta \frac{u m(\alpha )}{\alpha \gamma }} - v \right) . \end{aligned}$$

(13)

By setting $\epsilon = \frac{m(\alpha )}{r}$ as time scale parameter, we can write the dimensionless predator–prey system in slow time $\tau =\epsilon \hat{t}$:

$$\begin{aligned} \epsilon \frac{\text {d}x}{\text {d}x}&= u\left( 1-\frac{u m(\alpha )}{\gamma \alpha K}\right) - \frac{uv}{1+\beta \frac{u m(\alpha )}{\alpha \gamma }}\end{aligned}$$

(14)

$$\begin{aligned} \frac{\text {d}y}{\text {d}y}&= \left( \frac{uv}{1+\beta \frac{u m(\alpha )}{\alpha \gamma }} - v \right) . \end{aligned}$$

(15)

For $m(\alpha ) = \alpha ^2+c$, the separation between the prey’s and predator’s time scale is given by $\epsilon =\frac{(\alpha ^2+c)}{r}$. When $\alpha ^2 + c = r$, prey and predator evolve on the same time scale, otherwise when $\alpha ^2 + c \gtrless r$, the prey reproduces slower/faster than the predator does Fig. 9.

Fig. 9

For mortality rate $m(\alpha )=\alpha ^2+c$, the time scale separation $\epsilon$ increases with increasing attack rate $\alpha$. For $\alpha \gtrless 0.9$, the predator evolves faster/slower than its prey. For $\alpha \approx 0.9$, predator and prey evolve on the same time scale

The Eco-evolutionary system

In this section, we study the stability of the long-term states of the eco-evolutionary predator–prey system (16)–(18):

$$\begin{aligned} \frac{\text {d}x}{\text {d}x}&= rx\left( 1-\frac{x}{K(t)}\right) - \frac{\alpha xy}{1+\beta x}\end{aligned}$$

(16)

$$\begin{aligned} \frac{\text {d}y}{\text {d}y}&= \gamma \frac{\alpha x y}{1+\beta x} - m(\alpha )y\end{aligned}$$

(17)

$$\begin{aligned} \frac{\text {d}\alpha }{\text {d}\alpha }&= r_V(\alpha -\alpha _{min})(\alpha _{max}-\alpha )\left( \frac{\gamma x}{1+\beta x} - \frac{\text {d}m(\alpha )}{\text {d}m(\alpha )} \right) , \end{aligned}$$

(18)

with time-dependent parameter K(t): $K(t) = max(K_0-r_K t, K_{min})$. Equation (18) is zero if (i) $\alpha =\alpha _{min}$, (ii) $\alpha =\alpha _{max}$ or (iii) $\left( \frac{\gamma x}{1+\beta x} - \frac{\text {d}m(\alpha )}{\text {d}m(\alpha )} \right) =0$. With $m(\alpha )=\alpha ^2+c$, the eight equilibria $\varvec{e}_1(K(t),\epsilon )-\varvec{e}_8(K(t),\epsilon )$ (19)–(26) are given by:

$$\begin{aligned} \varvec{e}_1(K(t),\epsilon )&= (e_{1x}(K(t),\epsilon ), e_{1y}(K(t),\epsilon ), e_{1\alpha }(K(t),\epsilon )) = (0,0,\alpha _{min})\end{aligned}$$

(19)

$$\begin{aligned} \varvec{e}_2(K(t),\epsilon )&= (K(t),0,\alpha _{min}) \end{aligned}$$

(20)

$$\begin{aligned} \varvec{e}_3(K(t),\epsilon )&= \left( \frac{\alpha ^2_{min}+c}{\gamma \alpha _{min} - (\alpha ^2_{min}+c)\beta }, \frac{r}{\alpha _{min}}\left( 1-\frac{e_{3x}}{K(t)} \right) \left( 1-\beta e_{3x} \right) ,\alpha _{min} \right) .\end{aligned}$$

(21)

$$\begin{aligned} \varvec{e}_4(K(t),\epsilon )&= (0,0,\alpha _{max})\end{aligned}$$

(22)

$$\begin{aligned} \varvec{e}_5(K(t),\epsilon )&= (K(t),0,\alpha _{max}) \end{aligned}$$

(23)

$$\begin{aligned} \varvec{e}_6(K(t),\epsilon )&= \left( \frac{\alpha ^2_{max}+c}{\gamma \alpha _{max} - (\alpha ^2_{max}+c)\beta }, \frac{r}{\alpha _{max}}\left( 1-\frac{e_{6x}}{K(t)} \right) \left( 1-\beta e_{6x} \right) ,\alpha _{max} \right) \end{aligned}$$

(24)

$$\begin{aligned} \varvec{e}_7(K(t),\epsilon )&= \left( \frac{2c}{\gamma \sqrt{c} - 2c\beta }, \frac{r}{\sqrt{c}}\left( 1-\frac{e_{7x}}{K(t)} \right) \left( 1-\beta e_{7x} \right) ,\sqrt{c} \right) \end{aligned}$$

(25)

$$\begin{aligned} \varvec{e}_8(K(t),\epsilon )&= \left( K(t),0,\frac{\gamma K(t)}{2+2\beta K(t)}\right) \end{aligned}$$

(26)

Figure 10 shows the stability of all eight equilibria $\varvec{e}_1(K(t),\epsilon )-\varvec{e}_8(K(t),\epsilon )$ (19)–(26) depending on the time-dependent carrying capacity K(t) and the genetic variation $r_V$. The equilibria $\varvec{e}_1(K(t),\epsilon )$–$\varvec{e}_6(K(t),\epsilon )$ are unstable ((u), orange) for all values of K and $r_V$. Hence, the seventh $\varvec{e}_7(K(t),\epsilon )$ or eighth equilibrium $\varvec{e}_8(K(t),\epsilon )$ represents the globally stable equilibrium of the system depending on K(t). For higher genetic variation $r_V$, the stability range of the seventh equilibrium $\varvec{e}_7(K(t),\epsilon )$ extends even to higher values of K(t) (see Fig. 10g).

Fig. 10

Stability of the eight equilibria $\varvec{e}_1(K(t),\epsilon )$–$\varvec{e}_8(K(t),\epsilon )$ (19)–(26) depending on the genetic variance $r_V$ and the time-dependent carrying capacity K(t). Blue: stable (s), orange: unstable (u). Parameter: Table 1

Eco-evolutionary dynamics for different trade-offs

In this section, we study if R-tipping and indirect evolutionary rescue still occurs for different trade-offs between predation $\gamma \frac{\alpha x y}{1+\beta x}$ and the mortality of the predator $m(\alpha )y$. Therefore, we introduce the trade-off parameter k: $m(\alpha ) = \alpha ^k+c$ with $k \in \mathbf {R_0}^+$. The eco-evolutionary system becomes:

$$\begin{aligned} \frac{\text {d}x}{\text {d}x}&= rx\left( 1-\frac{x}{K(t)}\right) - \frac{\alpha xy}{1+\beta x}\end{aligned}$$

(27)

$$\begin{aligned} \frac{\text {d}y}{\text {d}y}&= \gamma \frac{\alpha x y}{1+\beta x} - m(\alpha )y\end{aligned}$$

(28)

$$\begin{aligned} \frac{\text {d}\alpha }{\text {d}\alpha }&= r_V(\alpha -\alpha _{min})(\alpha _{max}-\alpha )\left( \frac{\gamma x}{1+\beta x} - \frac{\text {d}m(\alpha )}{\text {d}m(\alpha )} \right) , \end{aligned}$$

(29)

with time-dependent parameter K(t): $K(t) = max(K_0-r_K t, K_{min})$. As shown by Eq. (28) and Eq. (29), the trade-off parameter k affects directly (i) the mortality of the predator $m(\alpha )$ and especially, (ii) the fitness gradient $\left( \frac{\gamma x}{1+\beta x} - \frac{\text {d}m(\alpha )}{\text {d}m(\alpha )} \right)$. Hence, the parameter k influences the equilibrium (optimum) state $\varvec{e}_{\text {opt}}(K(t),\epsilon )$ of the eco-evolutionary system (27)–(29) for which the fitness gradient vanishes $\left( \frac{\gamma x}{1+\beta x} - \frac{\text {d}m(\alpha )}{\text {d}m(\alpha )} \right) =0$.

Setting Eqs. (27)–(29) to zero and solving for the state variables x, y and $\alpha$ leads to the following eight equilibria $\varvec{e}_{1}(K(t),\epsilon )$–$\varvec{e}_{8}(K(t),\epsilon )$ (30)–(37):

$$\begin{aligned} \varvec{e}_{1}(K(t),\epsilon )&= (e_{1,x}(K(t),\epsilon ), e_{1,y}(K(t),\epsilon ), e_{1\alpha }(K(t),\epsilon )) = (0,0,\alpha _{min})\end{aligned}$$

(30)

$$\begin{aligned} \varvec{e}_{2}(K(t),\epsilon )&= (K(t),0,\alpha _{min}) \end{aligned}$$

(31)

$$\begin{aligned} \varvec{e}_{3}(K(t),\epsilon )&= \bigg ( \frac{\alpha ^k_{min}+c}{\gamma \alpha _{min} - (\alpha ^k_{min}+c)\beta },\nonumber \\&\frac{r}{\alpha _{min}}\left( 1-\frac{e_{3,x}}{K(t)} \right) \left( 1-\beta e_{3,x} \right) ,\alpha _{min} \bigg ) \end{aligned}$$

(32)

$$\begin{aligned} \varvec{e}_{4}(K(t),\epsilon )&= (0,0,\alpha _{max})\end{aligned}$$

(33)

$$\begin{aligned} \varvec{e}_{5}(K(t),\epsilon )&= (K(t),0,\alpha _{max}) \end{aligned}$$

(34)

$$\begin{aligned} \varvec{e}_{6}(K(t),\epsilon )&= \bigg ( \frac{\alpha ^k_{max}+c}{\gamma \alpha _{max} - (\alpha ^k_{max}+c)\beta }, \nonumber \\&\frac{r}{\alpha _{max}}\left( 1-\frac{e_{6,x}}{K(t)} \right) \left( 1-\beta e_{6,x} \right) ,\alpha _{max} \bigg )\end{aligned}$$

(35)

$$\begin{aligned} \varvec{e}_{7}(K(t),\epsilon )&= \bigg ( \frac{kc(k-1)^{-1}}{\gamma c(k-1)^{-1} - kc(k-1)^{-1}\beta }, \nonumber \\&\frac{r}{\left( \frac{c}{k-1}\right) ^{k^{-1}}}\left( 1-\frac{e_{7,x}}{K(t)} \right) \left( 1-\beta e_{7,x} \right) ,\left( \frac{c}{k-1}\right) ^{k^{-1}} \bigg )\end{aligned}$$

(36)

$$\begin{aligned} \varvec{e}_{8}(K(t),\epsilon )&= \left( K(t),0,\frac{\gamma K(t)}{2+2\beta K(t)}\right) . \end{aligned}$$

(37)

Notice setting $k=2$ leads to the optimum state $\varvec{e}_{\text {opt}}(K(t),\epsilon )$ (9) introduced in the section: Indirect evolutionary rescue prevents rate-induced tipping. There, the optimum state was represented by one of the two stable moving equilibria $\varvec{e}_{7}(K(t),\epsilon )$ and $\varvec{e}_{8}(K(t),\epsilon )$ depending on the value of K(t). Interestingly, for $k\ge 3$, $\varvec{e}_{7}(K(t),\epsilon )$ (36) represents the unique stable moving equilibrium for all values of the time-dependent carrying capacity K(t) because $\varvec{e}_{8}(K(t),\epsilon )$ (37) is unstable (compare the colored regions above $k\ge 3$ in Fig. 11). Hence, the predator population is not extinct in the long-term $\varvec{e}_{7}(K_{min},\epsilon )$ (36) ($e_{7,y}(K(t),\epsilon ) \ne 0$). Figure 12 demonstrates that we still find the rate-induced collapse of the prey and its prevention by indirect evolutionary rescue even though the predator persists in the long-term state $\varvec{e}_{7}(K_{min},\epsilon )$ (magenta dotted lines) for $k=3$.

Fig. 11

Stability of $\varvec{e}_{1}(K(t),\epsilon )$ (36) and $\varvec{e}_{8}(K(t),\epsilon )$ (37) depending on the trade-off parameter k and the ti0me-dependent carrying capacity K(t). Blue: stable (s), orange: unstable (u). Parameters: Table 1

Fig. 12

Prey density x, predator density y, attack rate $\alpha$ and predation pressure $pred = \frac{\alpha x y}{1+\beta x}$ of the eco-evolutionary system (27)–(29) for $r_V=0.1$ (left column) and $r_V = 0.9$ (right) column. The optimum state $\varvec{e}_{7}(K(t),\epsilon )$ (36) is marked as magenta dashed line. Parameter: $x_0 = n_x(e_{7,\alpha })$, $n_y(K_0=5.0,e_{7,\alpha } )$, $\alpha _0 = e_{7,\alpha }(K(t),\epsilon )$. Other parameters can be seen in Table 1