## Introduction

Nonlinear dynamics appears in a variety of fields, including classical mechanics, chemical reaction systems, and population biology, to name a few1. Nonlinearity can trigger complex temporal and spatial patterns and even chaotic behaviors, making it challenging to find universal relations within the properties of dynamics. In particular, slight perturbations in external parameters can result in qualitative changes in the dynamical property through a bifurcation such as the Hopf bifurcation, where self-sustained oscillation emerges. It is of pivotal importance to explore universal relations shared by a broad class of dynamical phenomena with nonlinearity.

Ecological and evolutionary processes often exhibit nonlinear population dynamics2,3 such as temporal oscillation in population sizes and irreversible extinction of certain species4. Typical biological systems consist of identifiable units such as genotypes and species (called “types” in this paper), and intra-type and inter-type interactions cause nonlinear dynamics2,4. Besides interactions, type-dependent growth rates determined by natural selection lead to nonlinear dynamics of the proportions of each type. In evolutionary theory, Fisher’s fundamental theorem of natural selection5,6 establishes a simple relation between the variance of the growth rate and the temporal increase in the average growth rate. The theorem has been extended to evolutionary models with mutation7,8 and ecological models9.

Bifurcations and associated critical dynamics play significant roles in biological processes10. In ecological11,12 and epidemiological13 systems, critical slowing down around bifurcation points has been discussed as an early warning signal for catastrophic shifts. In evolutionary systems, bifurcation points can appear as critical mutation rates beyond which heredity does not persist14,15, and the self-organized criticality has also been discussed as a possible mechanism of mass extinction of species16. Since such critical dynamics reflects instabilities behind nonlinear systems17, fundamental relations near bifurcation points are crucial in predicting dramatic changes in ecological and evolutionary processes.

We here derive a general constraint on nonlinear population dynamics by extending the formulation developed for stochastic processes18,19 to nonlinear dynamical systems. In particular, Fisher’s fundamental theorem of natural selection is a special case of the constraint. As a unique consequence of the constraint, we show that the critical scaling exponents of speeds near the bifurcation point should have nontrivial bounds that are universally determined by the type of bifurcation. We verify our theory for an evolutionary model with mutation and the susceptible-infected-recovered (SIR) model with birth and death, which show the transcritical bifurcation, as well as for the competitive Lotka–Volterra model, which undergoes limit-cycle oscillation.

## Results

### Constraint on general population dynamics

We consider a general population dynamics described by

$${\partial }_{t}{N}_{i}={F}_{i}({N}_{1},...,{N}_{L}),$$
(1)

where i is the label for each type, L is the total number of types, and Ni(t) is the density of type i at time t. If there are interactions between types, Fi(N1, . . . , NL) is generally a nonlinear function. Defining the proportion $$P:= {\{{P}_{i}\}}_{i = 1}^{L}:= {\{{N}_{i}/{N}_{{{{{{{{\rm{tot}}}}}}}}}\}}_{i = 1}^{L}$$ with the total population density $${N}_{{{{{{{{\rm{tot}}}}}}}}}:= \mathop{\sum }\nolimits_{i = 1}^{L}{N}_{i}$$, we obtain equations for Pi and Ntot as

$${\partial }_{t}{P}_{i}=\frac{{F}_{i}({N}_{{{{{{{{\rm{tot}}}}}}}}}{P}_{1},...,{N}_{{{{{{{{\rm{tot}}}}}}}}}{P}_{L})}{{N}_{{{{{{{{\rm{tot}}}}}}}}}}-{P}_{i}\mathop{\sum }\limits_{j=1}^{L}\frac{{F}_{j}({N}_{{{{{{{{\rm{tot}}}}}}}}}{P}_{1},...,{N}_{{{{{{{{\rm{tot}}}}}}}}}{P}_{L})}{{N}_{{{{{{{{\rm{tot}}}}}}}}}}$$
(2)

and $${\partial }_{t}{N}_{{{{{{{{\rm{tot}}}}}}}}}=\mathop{\sum }\nolimits_{i = 1}^{L}{F}_{i}({N}_{{{{{{{{\rm{tot}}}}}}}}}{P}_{1},...,{N}_{{{{{{{{\rm{tot}}}}}}}}}{P}_{L})$$, respectively. Even if Fi(N1, . . . , NL) is a linear function for all i, Eq. (2) can be a nonlinear equation, and bifurcations can occur as we discuss later.

Applying the Cauchy-Schwarz inequality to the Price equation20,21, which is derived from the conservation of the total proportion ($$\mathop{\sum }\nolimits_{i = 1}^{L}{P}_{i}=1$$), we obtain the speed-limit inequality (Supplementary Method 1):

$${v}_{A}\le {v}_{\lim }:= \sqrt{{I}_{{{{{{{{\rm{F}}}}}}}}}}:= \sqrt{\left\langle {({\partial }_{t}P/P)}^{2}\right\rangle }.$$
(3)

Note that an inequality whose expression is the same as Eq. (3) has been discussed for stochastic processes18,19, and the relation to the Price equation has been pointed out21. Here, we define the Fisher information IF22,23 and the speed $${v}_{A}:= | {\partial }_{t}\left\langle A\right\rangle -\langle {\partial }_{t}A\rangle | /{{\Delta }}A$$, which characterizes the temporal change rate of a type-dependent quantity $$A:= {\{{A}_{i}\}}_{i = 1}^{L}$$ that can depend on time in general [Supplementary Method 1, Fig. 1(a)]. Also, the average and standard deviation are defined as $$\left\langle A\right\rangle := \mathop{\sum }\nolimits_{i = 1}^{L}{P}_{i}{A}_{i}$$ and $${{\Delta }}A:= {(\langle {A}^{2}\rangle -{\left\langle A\right\rangle }^{2})}^{1/2}$$, respectively. The inequality [Eq. (3)] provides a universal upper bound on the speed of population dynamics, independent of the choice of quantity A [Fig. 1(b)]. We stress that Eq. (3) applies to nonlinear dynamics though the expression is equivalent to that for Markov processes18,19, where the probability distribution follows linear dynamics. For example, $${v}_{\lim }$$ in Eq. (3) can be a non-monotonic function of time, in contrast to Markovian relaxation processes, where $${v}_{\lim }$$ decays monotonically18. Note that Eq. (3) is different from the previously obtained speed-limit inequalities in nonlinear systems24,25, which have been discussed mainly for chemical reaction networks. Following Nicholson et al.19, we can interpret Eq. (3) as the uncertainty relation between the timescale of dynamical quantities ($${{v}_{A}}^{-1}$$) and the information of dynamics ($$\sqrt{{I}_{{{{{{{{\rm{F}}}}}}}}}}$$).

### Relation to Fisher’s fundamental theorem

Our general constraint includes Fisher’s fundamental theorem as a special case when applied to an evolutionary model with natural selection. We take Fi = siNi in Eq. (1), where si > 0 is the type-dependent growth rate. In such systems, Fisher’s fundamental theorem of natural selection asserts that the increase in the average growth rate is equal to the variance of the growth rate5,7, i.e., $${\partial }_{t}\left\langle s\right\rangle ={({{\Delta }}s)}^{2}$$. As shown in Supplementary Method 2, we find that Fisher’s fundamental theorem is a special case of Eq. (3), $${v}_{s}={v}_{\lim }$$. Note that $${v}_{\lim }$$ in Eq. (3) is equivalent to Crow’s index of opportunity for selection, which provides an empirical estimate of the maximum strength of natural selection acting on a given population26,27.

Furthermore, even when the growth rate depends on time and densities, we show that an extended version of the fundamental theorem6,9 is a special case of Eq. (3), where the equality in Eq. (3) is satisfied (Supplementary Method 2). Our result therefore covers a variety of previous results established in population biology in light of information theory and statistical physics. For more general dynamics with mutation, the speed-limit inequality [Eq. (3)] is satisfied for any quantity A, including the growth rate s, and thus regarded as a generalization of the fundamental theorem. For instance, if we take the typical length of type i as Ai (e.g., length of bacteria for several types of mutants), the average length can potentially change more quickly as the variance of the length is larger, according to Eq. (3). Note that other types of extensions of the fundamental theorem to evolutionary models with mutation has been formulated7,8.

### Speed limit for evolutionary dynamics

We next consider another evolutionary model with natural selection and mutation [Fig. 1(d)] by taking $${F}_{i}={s}_{i}{N}_{i}+\mathop{\sum }\nolimits_{j = 1}^{L}{m}_{ij}{N}_{j}$$ in Eq. (1)14,28. Here, si > 0 is the growth rate and mij ≥ 0 (i ≠ j) is the mutation rate from type j to i. To demonstrate the inequality [Eq. (3)], we take L = 3 with $${s}_{2}={s}_{3}=\bar{s}$$ and examine a situation where type 1 will survive (become extinct) after a long time if the growth rate $${s}_{1}=\bar{s}+r$$ is larger (smaller) than a critical value $$\bar{s}+{r}_{{{{{{{{\rm{c}}}}}}}}}$$ (Supplementary Method 3). The extinction transition at r = rc corresponds to the transcritical bifurcation1.

Figure 2(a) shows typical time dependence of the proportion Pi. As shown in Fig. 2(b–d), regardless of the value of r/rc, the speed of the growth rate vs (black solid lines) is bounded by the speed limit $${v}_{\lim }$$ (red dashed lines), which verifies Eq. (3). To confirm the generality of Eq. (3), we introduce the Shannon entropy $${I}_{{{{{{{{\rm{S}}}}}}}}}:= \left\langle I\right\rangle$$ with $$I:= {\{{I}_{i}\}}_{i = 1}^{L}:= {\{-\ln {P}_{i}\}}_{i = 1}^{L}$$23 as the (logarithm of) diversity of population (see Supplementary Fig. 1 for typical time dependence of IS). We show that the speed of change in diversity, vI (gray solid lines), is also bounded by $${v}_{\lim }$$.

### Universal constraint around transcritical bifurcation point

Let us examine a consequence of the speed limit at the transcritical bifurcation point (r = rc), where an observable A typically exhibits critical slowing down10,13 with a power-law decay of the speed, $${v}_{A} \sim {t}^{-{\alpha }_{A}}$$. While αA can vary for different A, the inequality [Eq. (3)] indicates that αA is bounded by a universal factor $${\alpha }_{\lim }$$ determined by the Fisher information [Fig. 1(c)]. Note that the power-law decrease in the Fisher information has also been discussed for the transient dynamics in nonlinear oscillator models29. In the evolutionary model with natural selection and mutation, we find P1 ~ t−1 (Supplementary Method 3) and thus

$${v}_{\lim } \sim \sqrt{{({\partial }_{t}{P}_{1})}^{2}/{P}_{1}} \sim {t}^{-{\alpha }_{\lim }^{{{{{{{{\rm{TC}}}}}}}}}}$$
(4)

with $${\alpha }_{\lim }^{{{{{{{{\rm{TC}}}}}}}}}=3/2$$. Then, we have

$${\alpha }_{A}\ge {\alpha }_{\lim }^{{{{{{{{\rm{TC}}}}}}}}}=3/2$$
(5)

for arbitrary A in this process.

In addition, if the parameter is slightly off the bifurcation point, the system can exhibit dynamical scaling, in a manner similar to critical phenomena30,31,32. Assuming that the relaxation times of the speed and the speed limit diverge at the bifurcation point as $$\sim \!| r-{r}_{{{{{{{{\rm{c}}}}}}}}}{| }^{-{\beta }_{A}}$$ and $$\sim \!| r-{r}_{{{{{{{{\rm{c}}}}}}}}}{| }^{-{\beta }_{\lim }^{{{{{{{{\rm{TC}}}}}}}}}}$$, respectively, we obtain the dynamical scaling laws as

$${v}_{A}(r-{r}_{{{{{{{{\rm{c}}}}}}}}},t)\simeq {t}^{-{\alpha }_{A}}{f}_{A}^{\pm }({t}^{1/{\beta }_{A}}| r-{r}_{{{{{{{{\rm{c}}}}}}}}}| ),$$
(6)
$${v}_{\lim }(r-{r}_{{{{{{{{\rm{c}}}}}}}}},t)\simeq {t}^{-{\alpha }_{\lim }^{{{{{{{{\rm{TC}}}}}}}}}}{f}_{\lim }^{\pm }({t}^{1/{\beta }_{\lim }^{{{{{{{{\rm{TC}}}}}}}}}}| r-{r}_{{{{{{{{\rm{c}}}}}}}}}| ),$$
(7)

where $${f}_{A}^{+}$$ and $${f}_{\lim }^{+}$$ ($${f}_{A}^{-}$$ and $${f}_{\lim }^{-}$$) are scaling functions for r − rc > 0 ( < 0). Combining the inequality [Eq. (3)] and the scaling laws [Eqs. (6) and (7)], we derive another constraint on the exponents as $${\beta }_{A}\le {\beta }_{\lim }^{{{{{{{{\rm{TC}}}}}}}}}$$ (Supplementary Method 3). In the numerical simulations, we have only found the case with $${\beta }_{A}={\beta }_{\lim }^{{{{{{{{\rm{TC}}}}}}}}}$$ (see below), which suggests that the diverging relaxation time of any speed should be proportional to the relaxation time of a single quantity (i.e., P1 in the present model) in a similar way to critical phenomena30,31.

To confirm the above argument, we demonstrate the long-time relaxation of $${v}_{\lim }$$, vs, vI, and a speed vb for the type index $$b:= {\{{b}_{i}\}}_{i = 1}^{L}:= {\{i\}}_{i = 1}^{L}$$ at the bifurcation point (r = rc) [Fig. 3(a)]. We find $${v}_{\lim } \sim {t}^{-3/2}$$ [red dotted line in Fig. 3(a)], which is consistent with Eq. (4). We also obtain vs ~ t−3/2, $${v}_{I} \sim {t}^{-2}\ln t$$, and vb ~ t−2 (see Supplementary Method 3 for the derivation), and the corresponding exponents are αs = 3/2, αI = 2 (neglecting the logarithmic dependence), and αb = 2, which indeed satisfy the inequality [Eq. (5)]. Moreover, slightly off the bifurcation point, we find the expected scaling laws [Eq. (6) and Eq. (7)] of vs, vb (Supplementary Fig. 2), and $${v}_{\lim }$$ [Fig. 3(b) and (c)] with $${\beta }_{s}={\beta }_{b}={\beta }_{\lim }^{{{{{{{{\rm{TC}}}}}}}}}=1$$.

Beyond specific dynamics, we conjecture that the exponents for the power-law decay of the speeds at the bifurcation point in population dynamics are bounded by a universal constant $${\alpha }_{\lim }$$ that only depends on the type of bifurcation. Similarly, the exponent $${\beta }_{\lim }$$ is also conjectured to be determined by the bifurcation type. These conjectures are plausible because critical properties associated with the bifurcation can be essentially described by the normal form for each bifurcation type1,32. This universal constraint on the exponents is a unique property of nonlinear dynamics, in contrast to the previous works on speed limits for linear dynamics18,19.

As a primary example, the inequality [Eq. (5)] can be generally applied to nonlinear dynamics that undergoes an extinction transition through the transcritical bifurcation. We consider the SIR model with birth and death [Fig. 1(e)], where N1, N2, and N3 are the densities of susceptible, infected, and recovered individuals, respectively33 (Supplementary Method 4). This model is genuinely nonlinear in that Fi(N1, N2, N3) in Eq. (1) is a nonlinear function. In this model, the transcritical bifurcation occurs as an extinction transition of the infected and recovered individuals, i.e., a transition between the disease-free and endemic states, and the critical slowing down occurs (P2 ~ P3 ~ t−1) at the bifurcation point (Supplementary Method 4). In Supplementary Fig. 3, we show typical time dependence of the proportion at the bifurcation point. We find that the speed of change in diversity vI and the speed limit $${v}_{\lim }$$ follow the same power-law decay as $${v}_{I} \sim {v}_{\lim } \sim {t}^{-3/2}$$ [Fig. 3(d)], satisfying the Eqs. (4) and (5).

### Universal constraint around Hopf bifurcation point

To verify our conjecture for other types of bifurcations, we focus on the Hopf bifurcation, at which a limit cycle starts to appear1. According to the normal form of the supercritical Hopf bifurcation, the deviation from the steady state decays with oscillation as $$\sim\!{t}^{-1/2}\cos \omega t$$ at the bifurcation point (Supplementary Method 5). Thus, for population dynamics undergoing the supercritical Hopf bifurcation, the proportion follows $${P}_{i} \sim {{{{{{{\rm{const.}}}}}}}}+{t}^{-1/2}\cos \omega t$$, and the speed limit decays as

$${v}_{\lim }=\sqrt{\mathop{\sum }\limits_{i=1}^{L}{({\partial }_{t}{P}_{i})}^{2}/{P}_{i}} \sim {t}^{-{\alpha }_{\lim }^{{{{{{{{\rm{Hopf}}}}}}}}}},$$
(8)

with $${\alpha }_{\lim }^{{{{{{{{\rm{Hopf}}}}}}}}}=1/2$$, where we only consider the amplitude relaxation by neglecting the oscillatory component. Correspondingly, if we assume a power-law decay of the speed amplitude as $${v}_{A} \sim {t}^{-{\alpha }_{A}}$$, αA should satisfy

$${\alpha }_{A}\ge {\alpha }_{\lim }^{{{{{{{{\rm{Hopf}}}}}}}}}=1/2.$$
(9)

As an ecological model that undergoes the supercritical Hopf bifurcation, we consider the competitive Lotka–Volterra model [Fig. 1(f)] by taking $${F}_{i}={s}_{i}{N}_{i}-\mathop{\sum }\nolimits_{j = 1}^{L}{c}_{ij}{N}_{i}{N}_{j}$$ in Eq. (1)4,34. Here, si is the growth rate, cij > 0 represents the competitive interaction between type i and j, and these parameters are set around the Hopf bifurcation (Supplementary Method 6).

We first show typical limit-cycle oscillation of the proportion [Fig. 4(a)]. Comparing vs, vI, and $${v}_{\lim }$$ within a single period [Fig. 4(b)], we confirm that the inequality [Eq. (3)] holds even when the limit cycle appears. By tuning the parameters to the Hopf bifurcation point, we numerically find the power-law decay of the speed amplitudes35 as $${v}_{s} \sim {v}_{I} \sim {v}_{\lim } \sim {t}^{-1/2}$$ [Fig. 4(c) and Supplementary Fig. 4], verifying Eqs. (8) and (9). Then, changing the parameters slightly off the bifurcation point, we find that the counterparts of the scaling laws [Eqs. (6) and (7)] hold for the speed amplitudes [Fig. 4(d) and Supplementary Fig. 5] with $${\beta }_{s}={\beta }_{I}={\beta }_{\lim }^{{{{{{{{\rm{Hopf}}}}}}}}}=1$$.

## Conclusion

We have illustrated the applications of the dynamical constraint [Eq. (3)] to ecological and evolutionary models. Focusing on the bifurcation unique to nonlinear dynamics, we have argued that the exponents of speeds at critical slowing down have the universal bounds that depend only on the bifurcation type. In particular, for the transcritical and supercritical Hopf bifurcations, we have confirmed the theoretically obtained Eqs. (4)–(9) using numerical simulations. Similar formulae are obtained for other bifurcations, e.g., $${\alpha }_{A}\ge {\alpha }_{\lim }^{{{{{{{{\rm{SN}}}}}}}}}=2$$ for the saddle-node bifurcation (Supplementary Method 7), which appears in population dynamics11,12,17.

Considering the probability18,19 instead of the proportion, we may extend our argument to critical phenomena in many-body stochastic systems, which can express nonequilibrium phenomena different from ecological and evolutionary dynamics. For instance, lattice gas models30, the contact process31, and biological systems such as swarms36 are potentially subject to constraints corresponding to Eqs (5) or (9) with possibly irrational lower bounds.

The methodologies of ecology and evolution have been developed almost independently37. However, ecological and evolutionary dynamics may not be separable in some situations. For example, rapid evolution can occur on the same timescale as that of ecological processes when there are drastic environmental changes37. General relations such as Eq. (3) will be useful in quantitative understanding of even inseparable eco-evolutionary dynamics.