Phase tipping: how cyclic ecosystems respond to contemporary climate

We identify the phase of a cycle as a new critical factor for tipping points (critical transitions) in cyclic systems subject to time-varying external conditions. As an example, we consider how contemporary climate variability induces tipping from a predator–prey cycle to extinction in two paradigmatic predator–prey models with an Allee effect. Our analysis of these examples uncovers a counterintuitive behaviour, which we call phase tipping or P-tipping, where tipping to extinction occurs only from certain phases of the cycle. To explain this behaviour, we combine global dynamics with set theory and introduce the concept of partial basin instability for attracting limit cycles. This concept provides a general framework to analyse and identify easily testable criteria for the occurrence of phase tipping in externally forced systems, and can be extended to more complicated attractors.


Introduction
Tipping points or critical transitions are fascinating nonlinear phenomena that are known to occur in complex systems subject to changing external conditions or external inputs. They are ubiquitous in nature and, in layman's terms, can be described as large, sudden and unexpected changes in the state of the system In the prey equation, −r(t)μ/ν is the low-density (negative) prey growth rate, cμ/ν quantifies the nonlinear modification of the low-density prey growth, the term (N − μ)/(ν + N) gives rise to the strong Allee effect that accounts for negative prey growth rate at low prey population density, α is the saturation predator kill rate and β is the predator kill half-saturation constant. The ratio r(t)/c is often referred to as the carrying capacity of the ecosystem. It is the maximum prey population that can be sustained by the environment in the absence of predators [44]. In the predator equation, χ represents the prey-to-predator conversion ratio and δ is the predator mortality rate. Realistic parameter values, estimated from Canada lynx and snowshoe hare data [43,44], can be found in table 1.
As we explain in §2(c), r(t) is a piecewise constant function of time that describes the varying climate. This choice makes the non-autonomous system (2.1) piecewise autonomous in the sense that it behaves like an autonomous system over finite time intervals. Therefore, much can be understood about the behaviour of the non-autonomous system (2.1) by looking at the autonomous frozen system with different but fixed-in-time values of r. The RMA frozen system can have at most four stationary states (equilibria), which are derived by settingṄ =Ṗ = 0 in (2.1). In addition to the extinction equilibrium e 0 , which is stable for r > 0, there is a prey-only equilibrium e 1 (r), the Allee equilibrium e 2 and the coexistence equilibrium e 3 (r), whose stability depends on r and other system parameters: e 0 = (0, 0), e 1 (r) = r c , 0 , e 2 = (μ, 0), e 3 (r) = (N 3 , P 3 (r)). (2.2) In the above, we include the argument (r) when an equilibrium's position depends on r. The prey and predator densities of the coexistence equilibrium e 3 (r) are given by:  The one-parameter bifurcation diagram of the RMA frozen system in figure 1a reveals various bifurcations and bistability, which are discussed in detail in §5(a) and appendix C. Most importantly, as r is increased, the coexistence equilibrium e 3 (r) undergoes a supercritical Hopf bifurcation H, which makes the equilibrium unstable and produces a stable limit cycle Γ (r). The cycle corresponds to oscillatory coexistence of predator and prey and is the main focus of this study. In the ecological literature, this Hopf bifurcation is referred to as the paradox of enrichment [68]. As r is increased even further, Γ (r) disappears in a dangerous heteroclinic bifurcation h at r = r h , giving rise to a discontinuity in the branch of coexistence attractors. Past r h , the only attractor is the extinction equilibrium e 0 . This heteroclinic bifurcation indicates where complete depletion of the predator becomes part of the cycle. Note that, in the absence of noise, the predator remains extinct once its level reaches zero because the subspace {P = 0} is invariant. Hence the counterintuitive transition to predator extinction at high prey growth rates.
To show that phase tipping is ubiquitous in predator-prey interactions, we also consider another paradigmatic predator-prey model, the May model [39,44]: This model has the same equation for the prey population density N as the RMA model, but differs in the equation for the predator population density P. Specifically, s is the low-density predator growth rate and is introduced to allow prey extinction. In other words, this model assumes that the predator must have access to other prey which allow it to survive at a low density /q in the absence of the primary prey N. The parameter q approximates the minimum preyto-predator biomass ratio that allows predator population growth, and table 1 contains realistic parameter values, estimated from Canada lynx and snowshoe hare data [43,44].
In addition to the extinction equilibrium e 0 , which is always stable, the May frozen system has a prey-only equilibrium e 1 (r), an Allee equilibrium e 2 , and two coexistence equilibria e 3 (r) and e 4 (r), whose stability depends on the system parameters. Further details and analysis of the May frozen model are provided in appendix A.    To depict phase tipping, each point on the limit cycle, as well as in a neighbourhood of the cycle, must be characterized by its unique phase. In the two-dimensional phase space of the autonomous predator-prey frozen systems (2.1) and (2.3), the stable limit cycle Γ (r) makes a simple rotation about the coexistence equilibrium e 3 (r). We take advantage of this fact and assign a unique phase ϕ γ ∈ [0, 2π ) to every point γ = (N γ , P γ ) on the limit cycle using a polar coordinate system anchored in e 3 (r) = (N 3 (r), P 3 (r)): In other words, the phase of the cycle is the angle measured counter-clockwise from the horizontal half line that extends from e 3 (r) in the direction of increasing N, as is shown in figure 2. Since the values of P(t) for the limit cycles in systems (2.1) and (2.3) are three orders of magnitude smaller than the values of N(t), the ensuing distribution of ϕ γ along Γ (r) is highly non-uniform. To address this issue and achieve a uniform distribution of ϕ γ , we include the factor of 10 3 in (2.4).
In the problem of P-tipping, we often encounter oscillatory solutions that have not converged to the limit cycle Γ (r). Equation (2.4) allows us to define the 'phase' of such oscillatory solutions in a neighbourhood of Γ (r).

(c) Climate variability
Climate variability here refers to changes in the state of the climate occurring on year-to-decade time scales. We model this process by allowing r(t), i.e. the prey birth rate and the carrying capacity of the ecosystem, to vary over time. This variation can be interpreted as climate-induced changes in resource availability or habitat quality. Seasonal modelling studies often assume sinusoidal variation in climate parameters [69][70][71][72], but many key climate variables vary much   more abruptly [41]. Since our unit of time is years, rather than months, we focus on abrupt changes in climate. 1 Guided by the approach proposed in [45,74], we construct a piecewise constant r(t) using two random processes; see figure 3a. First, we assume the amplitude of r(t) is a random variable with a continuous uniform probability distribution on a closed interval [r 2 , r 1 ]. Second, we assume the number of consecutive years during which the amplitude of r(t) remains constant is a random variable with a discrete probability distribution known as the geometric distribution 2 (2.5) where ∈ Z + is a positive integer and ρ ∈ (0, 1]. Such an r(t) can be viewed as bounded autocorrelated noise. Using actual climate records from four locations in the boreal and deciduous-boreal forest in North America, we choose a realistic value of ρ = 0.2 [45]. We say the years with constant r(t) are of high productivity, or Type-H, if their amplitude is greater than the mean (r 1 + r 2 )/2. Otherwise we say the years are of low productivity, or Type-L, as indicated in figure 3a.

B-tipping versus P-tipping in oscillatory predator-prey models
In this section, we use the non-autonomous RMA model (2.1) to demonstrate the occurrence of P-tipping in predator-prey interactions. Furthermore, we highlight the counterintuitive properties of P-tipping by a direct comparison with the intuitive and better understood B-tipping. Note that, in the non-autonomous system, e 0 remains the extinction equilibrium, but the predator-prey limit cycle Γ (r) is replaced by (irregular) predator-prey oscillations. Nonetheless, since the system is piecewise autonomous, the dynamics and bifurcations of the autonomous frozen system help us to understand the behaviour of the non-autonomous one.

(a) B-tipping from predator-prey cycles
We begin with a brief description of B-tipping due to the dangerous heteroclinic bifurcation h of the attracting predator-prey limit cycle Γ (r). In the autonomous frozen system, the cycle Γ (r) exists for the values of r below r h , and disappears in a discontinuous way when r = r h ; see figure 1a. Thus, we expect one obvious tipping behaviour in the non-autonomous system with a time-varying r(t): (B1) B-tipping from predator-prey oscillations to extinction e 0 will occur if r(t) increases past the dangerous bifurcation level r = r h , and the system converges to e 0 before switching back to r < r h . (B2) B-tipping will occur from all phases of predator-prey oscillations, but phases where the system spends more time are more likely to tip. An invariant measure μ(ϕ γ ) of Γ (r) can be obtained and normalized to approximate the probability distribution for B-tipping from a phase ϕ γ as shown in figure 3e; see ref. [76] and appendix B for more details on calculating μ(ϕ γ ). (B3) B-tipping from predator-prey oscillations cannot occur when r(t) decreases over time because Γ (r) does not undergo any dangerous bifurcations upon decreasing r.
To illustrate properties (B1)-(B3), we perform a Monte Carlo simulation of the non-autonomous RMA system (2.1). We restrict the variation of r(t) to the closed interval [r 2 , r 1 ] containing the bifurcation point r h (see the 'Climate variability' label in figure 1a, upper arrow), and perform 10 3 numerical experiments. In each experiment, we start from a fixed initial condition (N 0 , P 0 ) = (3, 0.002) within the basin of attraction of Γ (r), and let r(t) vary randomly as explained in §2c. We allow the system to continue until tipping from predator-prey oscillations to extinction occurs (figure 3b) due to a step change in r(t) from r pre to r post (figure 3a). We then record the values of r pre in red and the values of r post in blue in figure 3c, the state in the (N, P) phase space when the switch from r pre to r post occurs in figure 3d, and the corresponding phase of this state to produce the tipping-phase histograms in figure 3f. B-tipping is identified as the blue dots above  figure 3c, meaning that transitions to extinction occur when r(t) changes from r pre < r h to r post > r h in agreement with (B1) and (B3). The tipping phases corresponding to grey dots in figure 3d, and the ensuing grey histogram in figure 3f, correlate almost perfectly with the green invariant measure μ(ϕ γ ) of Γ (r) in figure 3e, in agreement with (B2).

(b) P-tipping from predator-prey cycles
The most striking result of the simulation is that B-tipping is not the only tipping mechanism at play. It turns out that there are other, unexpected and counterintuitive tipping transitions. These transitions indicate a new tipping mechanism, whose dynamical properties are in stark contrast to B-tipping: (P1) Tipping from the predator-prey oscillations to extinction occurs when r(t) decreases and does not cross any dangerous bifurcations of Γ (r), which is in contrast to (B1) and (B3). This is evidenced in figure 3c by the blue dots below r = r h depicting transitions to extinction when r(t) changes from r pre < r h to r post < r pre . (P2) Tipping occurs only from certain phases of predator-prey oscillations, which is in contrast to (B2). This is evidenced by the black dots in figure 3d, and the ensuing black tipping-phase histogram in figure 3f. (P3) The tipping phases do not correlate at all with the invariant measure μ(ϕ γ ) of Γ (r) shown in figure 3e. This is evidenced by a comparison with the black histogram in figure 3f.
Since the unexpected tipping transitions occur only from certain phases of predator-prey oscillations, we refer to this phenomenon as phase tipping or P-tipping.
Although P-tipping is less understood than B-tipping, it is ubiquitous and possibly even more relevant for predator-prey interactions. In figure 4, we restrict climate variability in the RMA model (2.1) to a closed interval [r 2 , r 1 ] that does not contain r h . In other words, we set r 1 < r h . Since the time-varying input r(t) cannot cross the dangerous heteroclinic bifurcation, all tipping transitions are P-tipping events. Furthermore, owing to the absence of dangerous bifurcations of Γ (r) in the May model (2.3) in figure 1b, P-tipping from predator-prey oscillations to extinction e 0 is the only tipping mechanism in figure 5. Note that P-tipping is more likely to occur in the May model, as evidenced by shorter tipping times; compare figures 4c and 5c.
The numerical experiments in figures 4 and 5 serve as motivating examples for the development of a general mathematical framework for P-tipping in §4.
(c) The Allee threshold: intuitive explanation of P-tipping Intuitively, P-tipping from predator-prey oscillations to extinction in the non-autonomous system can be understood in terms of an Allee threshold θ(r) in the autonomous frozen system, separating trajectories that lead to extinction from those that approach the predator-prey cycle (figures 2 and 10), and how a given drop in prey resources r(t) affects different phases near the predator-prey cycle via the changing Allee threshold.
The shape and position of both the Allee threshold θ(r) and the predator-prey cycle Γ (r) are modified by a drop in prey resources r(t). The strongest impact is expected when the drop coincides with the region of the fastest decline in prey N(t) and a large predator population P(t). These situations occur near the part of the cycle within a range of phases around ϕ γ = π/2, which is close to θ(r). There, the drop speeds up the prey decline, which, in conjunction with high predation pressure, creates perfect conditions for the ecosystem to move away from the modified cycle, cross the even closer modified Allee threshold and move towards extinction. Indeed, figures 4 and 5 show that P-tipping occurs from a range of phases around ϕ γ = π/2. The ecosystem response is very different if the same drop in prey resources coincides with the region of the fastest growth of prey N(t) and a small predator population P(t). These situations occur near a different part of the cycle, within a range of phases around ϕ γ = −π/2, which is away from θ(r). There, the Pr(tipping time = t) Pr(tipping phase = j g ) drop slows or even reverses the prey growth, but low predation pressure prevents the ecosystem from crossing the distant Allee threshold and helps it adapt to the modified cycle instead. Hence the observed phase sensitivity of tipping from predator-prey oscillations to extinction in the non-autonomous systems.

A geometric framework for P-tipping: partial basin instability
Motivated by the numerical experiments in figures 4 and 5, and the fact that P-tipping is not captured by classical bifurcation theory, the aim of this section is to provide mathematical tools for analysis of P-tipping. Specifically, we develop a simple geometric framework that uses global properties of the autonomous frozen system to study P-tipping from attracting limit cycles and their neighbourhoods in the non-autonomous system. The key concept is basin instability. 3 This concept was first introduced in ([13], Section 5.2) to study irreversible R-tipping from base states that are stationary (attracting equilibria) for fixed-in-time external inputs. Here, we extend this concept to base states that are attracting limit cycles for fixed-in-time external inputs. Our framework will allow us to give easily testable criteria for the occurrence of P-tipping from limit cycles in general, and explain the counterintuitive collapses to extinction in the predator-prey systems from §3. To define basin instability and P-tipping for limit cycles in general terms, we consider an n-dimensional non-autonomous systemẋ  with x ∈ R n , and a piecewise constant external input p(t) that can be single-switch or multi-switch. When it is important to highlight the dependence of multi-switch inputs on ρ (see equation (2.5)), we write p ρ (t) instead of p(t). Note that the RMA (2.1) and May (2.3) models with r(t) from §2(c) are examples of (4.1). Furthermore, we write to denote a solution to the non-autonomous system (4.1) at time t started from x 0 at initial time t 0 . We also consider the corresponding autonomous frozen system with different but fixed-in-time values of the external input p, and write to denote a solution to the autonomous frozen system at time t started from x 0 for a fixed p.

(a) Ingredients for defining basin instability
One key ingredient of a basin instability definition is the base attractor in the autonomous frozen system, denoted Γ (p), whose shape and position in the phase space vary with the input parameter(s) p. The second key ingredient is the basin of attraction of the base attractor, denoted B(Γ , p), whose shape and extent may also vary with the input parameter(s) p. For non-stationary attractors Γ (p), we work with the distance 4 between a solution x(t, x 0 ; p) and the set Γ (p), and x(t, x 0 , p) → Γ (p) as t → +∞, when this distance tends to zero as t → +∞. We define B (Γ , p) as the open set of all x 0 whose trajectories converge to Γ (p) forward in time: We often refer to the closure of the basin of attraction of Γ (p), denoted B(Γ , p), which comprises B(Γ , p) and its boundary, and to the basin boundary of Γ (p), which is given by the set difference B (Γ , p) \ B(Γ , p). Additionally, we assume that either all or part of the basin boundary of Γ (p) is a basin boundary of at least two attractors. This property, in turn, requires that the autonomous frozen system is at least bistable, meaning that it has at least one more attractor, other that Γ (p), for the same values of the input parameter(s) p.
The third key ingredient is a parameter path p , which we define as a connected set of all possible values of the external input p(t). It is important that p does not cross any classical autonomous bifurcations of the base attractor Γ (p).

(b) Definitions of basin instability for limit cycles
In short, basin instability of the base attractor on a parameter path describes the position of the base attractor at some points on the path relative to the position of its basin of attraction at other points on the path. Here, we define this concept rigorously for attracting limit cycles setwise. Definition 4.1. Consider a parameter path p . Suppose the frozen system has a family of hyperbolic attracting limit cycles Γ (p) that vary C 1 -smoothly with p ∈ p . We say Γ (p) is basin unstable on a path p if there are two points on the path, p 1 , p 2 ∈ p , such that the limit cycle Γ (p 1 ) is not contained in the basin of attraction of Γ (p 2 ): There exist p 1 , p 2 ∈ p such that Γ (p 1 ) ⊂ B(Γ , p 2 ). (4.2) Furthermore, we distinguish two observable (or typical) cases of basin instability: (i) We say Γ (p) is partially basin unstable on a path p if there are two points on the path, p 1 and p 2 ∈ p , such that the limit cycle Γ (p 1 ) is not fully contained in the closure of the basin of attraction of Γ (p 2 ), and, for every two points on the path, p 3 and p 4 ∈ p , Γ (p 3 ) has a non-empty intersection with the basin of attraction of Γ (p 4 ) and (ii) We say Γ (p) is totally basin unstable on a path p if there are (at least) two points on the path, p 1 and p 2 ∈ p , such that Γ (p 1 ) lies outside the closure of the basin of attraction of Γ (p 2 ) There exist p 1 , p 2 ∈ p such that Γ (p 1 ) B(Γ , p 2 ) = ∅. Additionally, there are two indiscernible (or special) cases of basin instability for limit cycles. They cannot be easily distinguished by observation from total basin instability, or from lack of basin instability. However, the indiscernible cases are necessary (although not sufficient) for the onset of partial basin instability and for transitions between partial and total basin instability. (iii) We say Γ (p) is marginally basin unstable on a path p if, in addition to (4.2), for every two points on the path, p 3 and p 4 ∈ p , the limit cycle Γ (p 3 ) is contained in B(Γ , p 4 ) Γ (p 3 ) ⊂ B(Γ , p 4 ) for every p 3 , p 4 ∈ p . (4.5) The special case of marginal basin instability separates the typical cases of 'no basin instability' and 'partial basin instability'. Furthermore, it is related to 'invisible R-tipping' and to transitions between 'tracking' and 'partial R-tipping' identified in [26]. (iv) We say Γ (p) is almost totally basin unstable on a path p if there are (at least) two points on the path, p 1 and p 2 ∈ p , such that Γ (p 1 ) does not intersect B(Γ , p 2 ), and, for every two points on the path, p 3 and p 4 ∈ p , the limit cycle Γ (p 3 ) intersects B(Γ , p 4 ).
There exist p 1 , p 2 ∈ p such that Γ (p 1 ) The special case of almost total basin instability separates the typical cases of 'partial basin instability' and 'total basin instability'. Furthermore, it is related to transitions between 'partial R-tipping' and 'total R-tipping' described in [26].
Note that, for equilibrium base states, 'partial basin instability' is not defined, whereas 'marginal basin instability' and 'almost total basin instability' become the same condition.
Guided by the approach proposed in [13], we would like to augment the classical autonomous bifurcation diagrams for the autonomous frozen system with information about (partial) basin instability of the base attractor Γ (p). The aim is to reveal non-autonomous instabilities that cannot be explained by classical autonomous bifurcations of the frozen system. To illustrate basin instability of Γ (p) in the bifurcation diagram of the autonomous frozen system, we define the region of basin instability of Γ (p) in the space of the input parameters as follows: Definition 4.3. In the autonomous frozen system, consider a C 1 -smooth family of hyperbolic attracting limit cycles Γ (p), and denote it with G. For a fixed p = p 1 , we define a region of basin instability of Γ (p 1 ) ∈ G as a set of all points p 2 in the space of the input parameters p, such that Γ (p 1 ) is not contained in the basin of attraction of Γ (p 2 ) ∈ G BI(Γ , p 1 ) := {p 2 : Γ (p 1 ) ⊂ B(Γ , p 2 ) and Γ (p 2 ) ∈ G}. (4.7)

(c) Partial basin instability and P-tipping
Thus far, we have worked with a loosely defined concept of P-tipping. In this section, we give rigorous definitions of P-tipping for single-switch and multi-switch p(t), show that partial basin instability of Γ (p) for a single-switch p(t) is necessary and sufficient for the occurrence of P-tipping from Γ (p), and discuss the applicability of this result to multi-switch p(t).

Definition 4.4.
Consider a non-autonomous system (4.1) with a piecewise constant input p(t) on a parameter path p . Suppose the autonomous frozen system has a family of hyperbolic attracting limit cycles Γ (p) that vary C 1 -smoothly with p ∈ p .
(i) Suppose p(t) is a single-switch that changes from p 1 ∈ p to p 2 ∈ p at time t = t 1 . Suppose also the system is on Γ (p 1 ) at t = t 1 . We then say that system (4.1) undergoes irreversible P-tipping from Γ (p 1 ) if there are x a , x b ∈ Γ (p 1 ), such that We call ϕ x b a tipping phase associated with each such x b . (ii) Suppose p ρ (t) is multi-switch with a fixed ρ. If x(t , x 0 ; t 0 ) leaves the basin of attraction B(Γ , p ρ (t)) for good, we use t 1 to denote the smallest switching time such that x(t, x 0 ; t 0 ) / ∈ B(Γ , p ρ (t)) for all t > t 1 .
We use x b = x(t 1 , x 0 ; t 0 ) to denote the corresponding state, and ϕ x b to denote the corresponding tipping phase. We then say that system (4.1) undergoes irreversible P-tipping if, for some initial condition x 0 ∈ B(Γ , p ρ (t 0 )) and all realizations of p ρ (t), there are tipping phases ϕ x b and also a non-zero Lebesgue measure subset of [0, 2π ) that does not contain any tipping phases ϕ x b .
We call t 1 the tipping time. (i) Smoothly varying p(t), for which P-tipping from Γ (p) is expected to depend on the rate of change of p(t) [26,30]. (ii) Non-periodic attractors such as tori or chaotic attractors, which may require an alternative phase definition. We return to this point in §6.
In general, the occurrence of P-tipping depends on the initial state, the properties of the external input p(t) and the topological structure of the phase space. We now show that partial basin instability of Γ (p) for a single-switch p(t) is necessary and sufficient for the occurrence of P-tipping from Γ (p).

Proposition 4.6.
Consider a non-autonomous system (4.1) and a parameter path p . Suppose the frozen system has a family of hyperbolic attracting limit cycles Γ (p) that vary C 1 -smoothly with p ∈ p , and Γ (p) is partially basin unstable on p . Then, for all p 1 and p 2 ∈ p , a single-switch parameter change from p 1 to p 2 gives irreversible P-tipping from Γ (p 1 ) if and only if Γ (p 1 ) ⊂ B(Γ , p 2 ).

Proof.
A single-switch parameter change from p 1 to p 2 at time t = t 0 reduces the problem to an autonomous initial value problem with initial condition x 0 = x(t 0 ) and fixed p = p 2 . It follows from the definition of basin of attraction that only solutions x(t, x 0 ; p 2 ) started from x 0 ∈ B(Γ , p 2 ) are attracted to the limit cycle Γ (p 2 ). Thus, if Γ (p) is partially basin unstable on p and Γ (p 1 ) ⊂ B(Γ , p 2 ), then there will be γ ∈ Γ (p 1 ) \ B(Γ , p 2 ) that give irreversible tipping, and γ ∈ Γ (p 1 ) ∩ B(Γ , p 2 ) that give no tipping. Conversely, if there is irreversible P-tipping from Γ (p 1 ), then there must be γ ∈ Γ (p 1 ) \ B(Γ , p 2 ), which implies Γ (p 1 ) ⊂ B(Γ , p 2 ). This rigorous statement no longer holds for multi-switch piecewise constant inputs p ρ (t). The reason is that trajectories are no longer guaranteed to converge to the limit cycle Γ (p), or to the alternative attractor of the frozen system, if the time interval between consecutive switches is short compared with the time of convergence. Additionally, trajectories started in the basin of attraction of Γ (p) may move away from Γ (p) for finite time. These differences allow for two dynamical scenarios that cannot occur in a system that starts on Γ (p) and is subject to a single-switch p(t).
In the first scenario, following a switch, the system leaves the basin of attraction of Γ (p), but fails to converge to an alternative attractor before the next switch happens, re-enters the basin of attraction of Γ (p) upon the second switch, and avoids P-tipping despite basin instability of Γ (p). We refer to such events as 'rescue events' [45]. Hence, basin instability of Γ (p) for a given switch within a multi-switch p(t) does not guarantee the occurrence of tipping upon this particular switch. For the second scenario, we extend the concept of partial basin instability to the whole basin of attraction of Γ (p). Suppose that Γ (p) is basin stable on p , but its basin of attraction is partially basin unstable on p . Following a switch, the trajectory moves away from Γ (p) and enters the basin unstable part of the basin of attraction of Γ (p), then the next switch happens, and the system undergoes P-tipping in the absence of basin instability of Γ (p). Hence, partial basin instability of Γ (p) need not be necessary for the occurrence of P-tipping with multi-switch p(t). Keeping in mind that multi-switch P-tipping is defined for all realizations of p ρ (t), it should be possible to show that, for multi-switch piecewise constant p(t): -Partial basin instability of Γ (p) on p is sufficient for the occurrence of P-tipping in system (4.1). -If p ρ (t) allows trajectories to converge to Γ (p) between all consecutive switches, then partial basin instability of Γ (p) on p is necessary and sufficient for the occurrence of P-tipping in system (4.1).

Partial basin instability and P-tipping in predator-prey models
In this section, we start with classical autonomous bifurcation analysis of the predator-prey frozen systems (2.1) and (2.3) to identify parameter regions with bistability between predatorprey cycles and extinction. Then, we show that predator-prey cycles can be partially basin unstable on several parameter paths r that lie within these regions of bistability. Finally, we demonstrate that partial basin instability of predator-prey cycles on a path r explains the counterintuitive collapses to extinction that occur only from certain phases of predatorprey oscillations, and gives simple testable criteria for the occurrence of P-tipping in the non-autonomous predator-prey system.

(a) Classical bifurcation analysis: limit cycles and bistability
There are four ecologically relevant parameter regions in the predator-prey frozen systems (2.1) and (2.3), shown in figure 6. These regions have qualitatively different dynamics that can be summarized in terms of stable states as follows: -Oscillatory Coexistence or Extinction. The system is bistable and can either settle at the extinction equilibrium e 0 , or self-oscillate as it converges to the predator-prey limit cycle Γ (r). Here is where P-tipping may occur; see the green regions in figure 6. -Stationary Coexistence or Extinction. The system is bistable and can settle either at the extinction equilibrium e 0 , or at the coexistence equilibrium e 3 (r); see the yellow regions in figure 6. -Prey Only or Extinction. The system is bistable and can settle either at the extinction equilibrium e 0 , or at the prey-only equilibrium e 1 (r); see the upper pink region in figure 6a. -Extinction. The system is monostable and can only settle at the extinction equilibrium e 0 ; see the other pink regions in figure 6.
The region boundaries are obtained via two-parameter bifurcation analysis using the numerical continuation software XPPAUT [78]. This analysis extends our discussion of the one-parameter bifurcation diagrams from figure 1. We refer to appendix C for the details of the bifurcation analysis, and to [79] for more details on classical autonomous bifurcation theory.

(b) Partial basin instability of predator-prey cycles
We now concentrate on the bistable regions labelled 'Oscillatory Coexistence or Extinction', apply definitions 4.1 and 4.3 to predator-prey cycles, and show that -Predator-prey cycles Γ (r) can be partially basin unstable on suitably chosen parameter paths. -Both predator-prey models have large parameter regions of partial basin instability.
When superimposed onto classical bifurcation diagrams, these regions reveal P-tipping instabilities that cannot be captured by the classical autonomous bifurcation analysis. -Partial basin instability of Γ (r) in the frozen system is sufficient for the occurrence of P-tipping in the non-autonomous system. The base attractor is the predator-prey limit cycle Γ (r), and the alternative attractor is the extinction equilibrium e 0 . The basin boundary of Γ (r) is the Allee threshold θ(r), which can be computed as the stable invariant manifold of the saddle equilibrium e s (r): θ(r) := W s (e s (r)) = {(N 0 , P 0 ) ∈ R 2 : (N(t), P(t)) → e s (r) as t → +∞}.
In the RMA frozen model, e s (r) is the saddle Allee equilibrium e 2 , whereas in the May frozen model, e s (r) is the saddle coexistence equilibrium e 4 (r) that lies near the repelling Allee equilibrium e 2 . To uncover the full extent of partial basin instability for the predator-prey cycles Γ (r), we fix a point p 1 that lies within the region labelled 'Oscillatory Coexistence or Extinction'; see figures 7a and 8a. Then, we apply definition (4.7) to identify all points p 2 within this region such that the predator-prey limit cycle Γ (p 1 ) is not contained in the closure of the basin of attraction of Γ (p 2 ). The ensuing (light grey) regions of partial basin instability bounded by the (dark grey) curves of marginal basin instability are superimposed on the classical bifurcation diagrams in figures 7a and 8a. Note that the basin instability regions BI(Γ , p 1 ) depend on the choice of p 1 , and are labelled simply BI for brevity. To illustrate the underlying mechanism in the (N, P) phase plane, we restrict to parameter paths r that are straight horizontal lines from p 1 in the direction of decreasing r. In other words, we set p = r; see figures 7a and 8a. When r 2 ∈ r lies on the dark grey curve of marginal basin instability, there is a single point of tangency between Γ (r 1 ) and θ (r 2 ), denoted γ ± in figures 7d and 8d. When r 2 ∈ r lies within the light grey region of partial basin instability, there are two points of intersection between Γ (r 1 ) and θ(r 2 ), denoted γ − and γ + in figures 7e and 8e. These two points bound the (red) part of the cycle that is basin unstable. The corresponding basin unstable phases are shown in figures 7b and 8b. Suppose that r(t) = r 1 , and a trajectory of the non-autonomous system is on the same side of θ(r 2 ) as the (red) basin unstable part of Γ (r 1 ). Then, when r(t) changes from r 1 to r 2 , the trajectory finds itself in the basin of attraction of the extinction equilibrium e 0 , and will thus approach e 0 .
The striking similarity is that predator-prey cycles from both models exhibit partial basin instability upon decreasing r. This decrease corresponds to climate-induced decline in the resources or in the quality of habitat. Furthermore, while the predator-prey cycle in the May model has a noticeably wider range of basin unstable phases, neither cycle appears to be totally basin unstable. All these observations are consistent with the counterintuitive properties (P1)-(P3) of P-tipping identified in the numerical experiments in §3.

(c) Partial basin instability explains P-tipping
Now, we can demonstrate that partial basin instability of Γ (r) in the autonomous predator-prey frozen systems explains and gives simple testable criteria for the occurrence of P-tipping in the non-autonomous systems. The families of attracting predator-prey limit cycles Γ (r), and their basin boundaries θ (r), are the two crucial components of the discussion below. First, recall the numerical P-tipping experiments from §3, and focus on the crescent-shaped 'clouds' of states from which P-tipping occurs; see the black dots in figures 4 and 5. Second, recognize that each P-tipping event occurs for a different value of r pre ∈ [r 2 , r 1 ], and thus from a different predator-prey cycle Γ (r pre ) or its neighbourhood. Therefore, we must consider the union of all cycles from the family along the parameter path r bounded by r 2 and r 1 : which is shown in figure 9. Furthermore, we use the basin boundary θ(r 2 ) of the cycle Γ (r 2 ) at the left end of the path to divide G into its (light green) basin stable part and (pink) basin unstable part on r with r ∈ [r 2 , r 1 ]. The 'clouds' of states from which P-tipping occurs agree perfectly with the basin unstable part of G. A few black dots that lie slightly outside the basin unstable part of G in figure 9b correspond to those P-tipping events that occur from states that have not converged to the limit cycle Γ (r pre ) and lie visibly away from Γ (r pre ) when the switch that causes tipping  happens. Those P-tipping events occur if the time interval during which r(t) = r pre is shorter than the time of convergence to the limit cycle Γ (r pre ) in the autonomous frozen system. For this particular parameter path, we could not detect any tipping events in the absence of partial basin instability of Γ (r). However, we could detect multiple 'rescue events' described in §4(c) (not shown in the figure). In a 'rescue event', the system leaves the basin of attraction of the predator-prey cycle after a switch that gives basin instability, but avoids tipping upon this switch because it re-enters the basin of attraction of the predator-prey cycle after some future switch. 'Rescue events' occur if the time interval during which r(t) = r pre is shorter than the time of convergence to the extinction equilibrium e 0 in the autonomous frozen system. In summary, the general concept of partial basin instability of Γ (r) on a parameter path r from definition 4.1 is an excellent indicator for the occurrence of P-tipping in the RMA (2.1) and May (2.3) models.

Conclusion
This paper studies nonlinear tipping phenomena, or critical transitions, in non-autonomous dynamical systems with time-varying external inputs. In addition to the well-known critical factors for tipping in systems that are stationary in the absence of external inputs, namely bifurcation, rate of change and noise, we identify here the phase of predator-prey limit cycles and nearby oscillations as a new critical factor in systems that are cyclic in the absence of external inputs.

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royalsocietypublishing.   To illustrate the new tipping phenomenon in a realistic setting, we consider two paradigmatic predator-prey models with an Allee effect, namely the RMA model [38] and the May model [39]. We describe temporal changes in the carrying capacity of the ecosystem with real climate variability records from different communities in the boreal and deciduous-boreal forest [45], and use realistic parameter values for the Canada lynx and snowshoe hare system [43,44]. Monte Carlo simulation reveals a robust phenomenon, where a drop in the carrying capacity tips the ecosystem from predator-prey oscillations to extinction. The special and somewhat counterintuitive result is that tipping occurs: (i) without crossing any bifurcations, and (ii) only from certain phases of the oscillations. Thus, we refer to this phenomenon as phase tipping (partial tipping), or simply Ptipping. Intuitively, P-tipping from predator-prey oscillations to extinction arises because a fixed drop in prey resources has distinctively different effects when applied during the phases of the oscillations with the fastest growth and the fastest decline of prey.
Motivated by the outcome of the simulation, we develop an accessible and general mathematical framework to analyse P-tipping and reveal the underlying dynamical mechanism. Specifically, we employ notions from set-valued dynamics to extend the geometric concept of basin instability, introduced in [13] for equilibria, to limit cycles. The main idea is to consider the autonomous frozen system with different but fixed-in-time values of the external input along some parameter path, and examine the position of the limit cycle at some point on the path relative to the position of its basin of attraction at other points on the path. First, we define different types of basin instability for limit cycles, and focus on partial basin instability that does not exist for equilibria. Second, we show that partial basin instability in the autonomous frozen system is necessary and sufficient for the occurrence of P-tipping in the non-autonomous system with a single-switch external input. Furthermore, we discuss the applicability of this result to multi-switch external inputs. Third, we relate our results to those of ref. [26] on rate-induced tipping from limit cycles.
We then apply the general framework to the ecosystem models and explain the counterintuitive transitions from certain phases of predator-prey oscillations to extinction. We use classical autonomous bifurcation analysis to identify parameter regions with bistability between predator-prey cycles and extinction. In this way, we show that predator-prey cycles can be partially basin unstable on typical parameter paths within these bistability regions. Moreover, we superimpose regions of partial basin instability onto classical autonomous bifurcation diagrams to reveal P-tipping instabilities that are robust but cannot be captured by classical bifurcation analysis.
We believe that this approach will enable scientists to uncover P-tipping in many different cyclic systems from applications ranging from natural science and engineering to economics. For example, the predator-prey paradigm is found across biological applications modelling, including epidemiology [80], pest control [81], fisheries [82], cancer [83,84] and agriculture [85,86]. The fundamental relationship described in predator-prey models also appears in many areas outside of the biological sciences, with recent examples including atmospheric sciences [87], economic development [64,65], trade and financial crises [88][89][90] and land management [91]. External disturbances of different kinds exist in all of these systems, suggesting that the P-tipping behaviours discovered in this paper are of broad practical relevance.
Furthermore, the concept of P-tipping, for base states that are attracting limit cycles with regular basin boundaries, naturally extends to more complicated base states, such as quasiperiodic tori and chaotic attractors, and to irregular (e.g. fractal) basin boundaries [28,31,32,92]. Defining phase for more complicated cycles in higher dimensions, and for non-periodic oscillations, will usually require a different approach. For example, one could define phase for an attracting limit cycle in a multi-dimensional system in terms of its period T as a linear function of time ϕ = 2π t/T. This definition is independent of the coordinate system, and can be extended to every point in the basin of attraction using isochrones. Another approach is to work with a time series of a single observable and use the Hilbert transform to construct the complex-valued analytic signal, and then extract the so-called instantaneous phase [93,94]. This phase variable may provide valuable physical insights into the problem of P-tipping when the polar coordinate approach does not work, or when the base attractor or its basin boundary have complicated geometry and are difficult to visualize. Such systems will likely exhibit even more counterintuitive tipping behaviours, but their analysis requires mathematical techniques beyond the scope of this paper.
Another interesting research question is that of early warning indicators for P-tipping. In the past decade, many studies of noisy real-world time-series records revealed prompt changes in the statistical properties of the data prior to tipping [1,21,22,95], which appear to be generic for tipping from equilibria. However, it is unclear if these statistical early warning indicators appear for P-tipping, or if one needs to identify alternatives such as finite time Lyapunov Exponent [96].
The one-parameter bifurcation diagram of the May frozen system in figure 1b reveals different bifurcations and bistability. Most importantly, as r is increased, the coexistence equilibrium e 3 (r) gives rise to a stable limit cycle Γ (r) via a safe supercritical Hopf bifurcation, denoted H 1 . The cycle exists for a range of r, and disappears in a reverse supercritical Hopf bifurcation, denoted H 2 , for larger r (see table 1).