Frequency-induced Tipping: Resonance-Driven Tipping in a Threshold Predator-Prey System

Introduction

Tipping, or a critical transition, refers to sudden, significant, and often difficult-to-reverse changes in the state of a dynamical system that can be triggered by small changes or perturbations[1, 2]. Such transitions are widely documented in ecological systems, where nonlinear feedbacks and alternative attractors can cause ecosystems to respond gradually over some ranges of environmental change but suddenly shift once resilience is sufficiently eroded[2]. Classic ecological tipping examples include eutrophication-driven transitions in shallow lakes from clear, macrophyte-dominated states to turbid, phytoplankton-dominated states; vegetation collapse and desertification in drylands; and coral reef phase shifts from coral-dominated to macroalgae-dominated communities[2, 3, 4, 5]. Understanding the dynamical mechanisms that generate such tipping phenomena is central to assessing resilience, predicting ecosystem collapse, and designing effective management strategies.

Although critical transitions, by themselves, may appear phenomenologically similar, the mechanisms that generate them can differ substantially[6, 7]. In the context of mathematical theory, four major mechanistic scenarios have been proposed to explain critical transitions in dynamical systems. Previous studies have often focused on the sudden shifts that occur when gradual quantitative changes in system conditions (i.e. parameters) alter the underlying stability structure. This mechanism is referred to as bifurcation-induced tipping or B-tipping[2, 8, 9]. A classic example is hysteresis, where the system shifts from a base attractor to an alternative stable state as an external condition gradually crosses a bifurcation threshold. Once this transition has occurred, simply reversing the external condition may not restore the original state; instead, a larger decrease or increase in the parameter is required to push the system back to its base attractor; see Figure 1A.

Crossing a bifurcation threshold, however, is not necessary for tipping. For instance, stochastic disturbances can perturb the system state or its parameters, driving a trajectory across a basin boundary into the domain of attraction of an alternative regime. If such disturbances cause the system to deviate sufficiently toward a different basin of attraction, the system can shift to another attractor, as illustrated in Figure 1B. This phenomenon is known as noise-induced tipping (N-tipping)[10, 11]. In contrast to B-tipping, which requires sufficient qualitative changes, the rate of change in the parameter can also induce rapid shifts in system states, referred to as rate-induced tipping (R-tipping)[12, 13]. R-tipping occurs when the system fails to track a moving attractor under sufficiently rapid parameter change, often without crossing a bifurcation threshold (Figure 1C). For systems exhibiting stable limit-cycles, phase-induced tipping (P-tipping) has been proposed as a means to destabilise the dynamics[14, 15]. P-tipping occurs when a parameter shift displaces a system from a limit-cycle into an alternative basin of attraction during the vulnerable phase of the cycle, even though the shift does not exceed the bifurcation threshold (Figure 1D).

Tipping mechanisms described above distinguish several ways in which the temporal structure and stochasticity of external forcing can affect system stability. Yet they do not fully address a common feature of environmental variability: repeated forcing at a characteristic frequency. Many environmental drivers fluctuate periodically or quasi-periodically across a wide range of temporal scales[16]. At short timescales, diel cycles in light and temperature entrain circadian regulation and shape daily physiological and behavioral rhythms in many organisms[17, 18]; intermediate-scale cycles include seasonal variation in temperature, precipitation, resource availability, and life-history timing[19]; and longer-timescale variability includes climate modes such as the El Niño–Southern Oscillation (ENSO), the North Atlantic Oscillation (NAO), and the Pacific Decadal Oscillation (PDO), whose periods can span years and influence ecological processes across broad spatial scales[20]. These examples suggest that environmental change is often structured not only by its magnitude or direction, but also by its characteristic frequency.

Periodic forcing can alter both the occurrence and interpretation of tipping in non-autonomous systems[21, 22]. In such systems, tipping may involve transitions between time-dependent attracting responses rather than between static equilibria, and tipping occurrence can depend on the amplitude, period, frequency, and phase of the external forcing[15]. Among the properties of periodicity, resonance provides a natural mechanism for this frequency dependence. For instance, periodic environmental variation can be amplified by nonlinear ecological responses, producing resonance-like population outbreaks[23], while stochastic resonance has been shown to arise near a critical transition in a lake-eutrophication model[24]. Stochastic resonance has also been linked to increased phase-tipping risk in cyclic predator–prey systems under climate variability[15]. These results imply that environmental fluctuations with the same mean and amplitude may have qualitatively different effects at different frequencies. This motivates the following hypothesis:

Subthreshold forcing can induce tipping when applied at a resonant frequency.

In this paper, we identify and characterize frequency-induced tipping (F-tipping): tipping occurs when external periodic forcing resonates with a system’s intrinsic dynamics and amplifies endogenous variation enough to drive the trajectory out of the basin of attraction of its base attractor, although that regime remains stable for every frozen forcing value within the forcing range. We compare F-tipping with other tipping mechanisms in Table 1.

Existing mechanism	Primary trigger	Distinction from F-tipping
B-tipping	A slowly varying parameter crosses a bifurcation threshold at which the original attractor loses stability or disappears.	F-tipping can occur even when the periodic forcing remains within bounds for which the system does not tip under fixed parameter values.
N-tipping	Random fluctuations push the state or parameters across a basin boundary or critical threshold.	F-tipping is driven by deterministic periodic forcing rather than stochastic perturbations. The transition depends on forcing frequency, not noise amplitude or rare random events.
R-tipping	Non-autonomous instability in which the system loses the ability to track a changing base attractor, without crossing a frozen-system bifurcation.	F-tipping emphasizes resonant amplification at selected forcing frequencies, rather than tracking failure alone.
P-tipping	Perturbations applied at vulnerable phases of an existing oscillation drive the system away from its original attractor.	F-tipping does not require a pre-existing stable limit cycle or perturbation at a dangerous phase. Its defining feature is frequency-selective amplification, although the external forcing phase can also affect the outcome.

This perspective has direct implications for ecosystem management, especially in systems whose dynamics are governed by intrinsic timescales such as growth, mortality, dispersal, recovery, resource renewal, and species interactions. Management strategies often focus on reducing the magnitude of environmental stressors, slowing their rate of change, or preventing extreme perturbations[1, 6, 8, 13]. F-tipping suggests an additional control dimension: the recurrence frequency of environmental variation. Identifying frequencies that resonate with intrinsic ecological timescales could help managers avoid destabilizing intervention schedules, anticipate periods of heightened vulnerability, and design restoration or disturbance regimes that minimize resonant amplification. Preventing or promoting transitions may require not only asking how much, how fast, or when the environment changes, but also how often it changes.

Results

Filippov predator-prey model

We first introduce the autonomous predator-prey system that provides the unforced baseline for the frequency-induced tipping analysis:

with the Heaviside step function:

In this model, $N(t)$ and $P(t)$ denote the prey and predator populations at time $t$. The prey dynamics are governed by their intrinsic growth rate $r$, carrying capacity $K$, and Allee effect threshold $N_a$. For predators, parameters $a$, $b$, $m$, and $P_a$ denote the predator attack rate, conversion efficiency, natural mortality rate, and the predator Allee effect threshold, respectively.

The Heaviside functions $\theta(\cdot)$ introduce abundance-dependent switches in prey growth and predation, representing idealized Allee effects in both populations. For prey, the intrinsic population dynamics follow logistic growth when $N>N_a$ and decline exponentially otherwise. This switch captures demographic disadvantages at low abundance, such as mate limitation, reduced group vigilance, and weaker collective defence[25, 26]. $N_a$ therefore represents the threshold below which the prey population cannot sustain itself even in the absence of predation. For predators, $P_a$ represents the minimum abundance required for effective predation, as might arise from dependence on cooperative hunting. When $P\leq P_a$, predation is inactive and predators decline solely through natural mortality. When $P>P_a$, predation removes prey at rate $aNP$ and contributes to predator reproduction at rate $baNP$. These switches make system (1) piecewise-smooth, with discontinuities along $N=N_a$ and $P=P_a$, and motivate its treatment within the Filippov framework[27].

Assuming $N_a>0$ and $P_a>0$, these thresholds partition the nonnegative phase plane $\mathbb{R}^{2}_{+}:=\{(x,y):x\geq0,y\geq0\}$ into four distinct regions $R_i$, $i\in\{1,2,3,4\}$, as shown in Figure 2A. Within each region, system (1) is governed by a smooth vector field $f_i$. The boundaries separating adjacent regions are denoted by $\Sigma_{(i,j)}$, with $(i,j)\in\{(1,2),(2,3),(3,4),(4,1)\}$. The vector field therefore switches whenever a trajectory intersects one of these boundaries. The full subsystem formulation and formal Filippov definitions are detailed in Methods.

Unless otherwise specified, we use the baseline parameters in (11). The unforced system (1) has three biologically relevant candidate equilibria:

corresponding to extinction, prey-only persistence, and predator-prey coexistence, respectively. Note that the coexistence equilibrium $E_2$ is always regular ($N^* > N_a$ and $P^* > P_a$, see Definition1) under our parameter choices. We numerically computed basins of attraction, as shown in Figure 2B, with representative simulations illustrated in Figure 2C-E. The basins of attraction show that if the trajectory crosses the prey threshold $N_a$, the dynamics can lead to extinction $E_0$; if it crosses the predator threshold $P_a$ while prey remain above its Allee threshold, the dynamics can lead to prey-only persistence $E_1$. Stability analyses are provided in the Supplementary Material C. Specifically, we compute the eigenvalues of the Jacobian matrix of subsystem $f_1$ evaluated at $E_2$

Since $\Re(\lambda_{\pm}) < 0$ and $\Im(\lambda_{\pm}) \ne0$ for appropriate parameter choices (see Supplemental Material A), the coexistence equilibrium $E_2 = (N^*,P^*)$ is a stable focus. The natural period of the associated damped oscillations is $\tau_n = \frac{2\pi}{\omega}$. This intrinsic period motivates the mechanism studied below: periodic forcing of the prey growth rate can resonate with the damped predator-prey oscillations when the forcing period is close to $\tau_n$, amplifying the trajectory such that a switching boundary is reached, triggering a critical transition.

Periodic forcing induces frequency-selective transition

Environmental variation, including seasonal changes, can cause temporal variation in demographic rates[28], which is commonly represented using sinusoidal forcing[29]. We model the periodically varying intrinsic growth rate as

where $r_0$ is the mean growth rate, $A$ is the forcing amplitude, $\tau$ is the forcing period, and $\phi$ is the initial phase. Here $0 \le A < r_0$, ensuring that $\rho(t)$ stays positive. Replacing the constant growth rate $r$ in system (1) with $\rho(t)$ gives

Here we choose $r_0=1.2$ and $\phi=0$, and initialise the system at the coexistence equilibrium $E_2(r_0)$. For these parameters, the frozen system admits a coexistence equilibrium for $r > r_c \approx0.978484$ (Methods 4.1.3), and no frozen-system bifurcation threshold was crossed since $\min_t \rho(t)=r_0-A>r_c$.

To visualize the resonant response before accounting for threshold switching, we simulate the smooth subsystem $f_1$ in $R_1$ with $\rho(t)$. For each fixed forcing amplitude $A \in\{0,0.05,0.10,0.15\}$, we vary the forcing period over $\tau\in[0.05,20]$ and record the historical minimum and maximum populations over $t \in[0,250]$ (Figure 3A,B). All amplitudes satisfied the above condition, with the largest amplitude giving $\min_t \rho(t)=1.05>r_c$. Results show pronounced amplification for forcing periods $\tau$ near the natural period of the unforced coexistence equilibrium $\tau_n=\frac{2\pi}{\omega}$. Increasing $A$ raises the historical maximum and lowers the historical minimum, widening the range of forcing periods over which the prey or predator minimum fell below its corresponding threshold $N_a$ or $P_a$. In the full piecewise system, such crossings activate different dynamical regimes and can lead to population collapse.