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In this paper, we study extinction in dynamical systems generated by reaction networks. We introduce two notions: weak extinction and strong extinction, and relate them to the structure of the underlying network through Lyapunov functions and LaSalle's invariance principle. In particular, for all deficiency-zero networks that are not weakly reversible, we provide a geometric construction of linear Lyapunov functions. Using these functions, we establish that if these networks have bounded invariant subspaces, then they must exhibit weak extinction within every such subspace. Also, for linear networks that are not weakly reversible, we show that every species outside a terminal strongly connected component undergoes strong extinction. Moreover, in order to further emphasize the difference between weak and strong extinction, we construct an example of a reaction system (based on the Ivanova network) that exhibits weak extinction for all the species, but does not exhibit strong extinction in any species.

Studies that predict species extinction have focused on a range of flora and fauna but in regard to Homo sapiens there are, with one notable exception, no predictive studies, only considerations of possible ways this may occur. The exception believes extinction of Homo sapiens will happen in 10,000 years. We agree that extinction will happen, but we disagree on the timing: The work we present here suggests that if the current decline in birth rates continues, humans could be extinct by 2394. If we consider the absence of working-age people and the accompanying collapse of services, the survivorship rates would most likely be lower. Given this, it is plausible that extinction could occur around 2359. We also examined a scenario in which births ended in 2024, which revealed that Homo sapiens would become extinct in 2134. Given societal collapse, extinction under the zero births scenario could occur around 2089.

Multispecies ecosystems modelled by generalized Lotka-Volterra equations exhibit stationary population abundances, where large number of species often coexist. Understanding the precise conditions under which this is at all feasible and what triggers species extinctions is a key, outstanding problem in theoretical ecology. Using standard methods of random matrix theory, I show that distributions of species abundances are Gaussian at equilibrium, in the weakly interacting regime. One consequence is that feasibility is generically broken before stability, for large enough number of species. I further derive an analytical expression for the probability that $n=0,1,2,...$ species go extinct and conjecture that a single-parameter scaling law governs species extinctions. These results are corroborated by numerical simulations in a wide range of system parameters.

Mass extinction is a phenomenon in the history of life on Earth when a considerable number of species go extinct over a relatively short period of time. The magnitude of extinction varies between the events, the most well known are the ``Big Five'' when more than one half of all species got extinct. There were many extinctions with a smaller magnitude too. It is widely believed that the common trigger leading to a mass extinction is a climate change such a global warming or global cooling. There are, however, many open questions with regard to the effect and potential importance of specific factors and processes. In this paper, we develop a novel mathematical model that takes into account two factors largely overlooked in the mass extinctions literature, namely, (i) the active feedback of phytoplankton to the climate through changing the albedo of the ocean surface and (ii) the species's adaptive evolutionary response to a climate change. We show that whether species goes extinct or not depends on a subtle interplay between the scale of the climate change and the rate of the evolutionary response. We also show that species's response to a fast climate change can exhibit long transi

Given a branching random walk on a set $X$, we study its extinction probability vectors $\mathbf q(\cdot,A)$. Their components are the probability that the process goes extinct in a fixed $A\subseteq X$, when starting from a vertex $x\in X$. The set of extinction probability vectors (obtained letting $A$ vary among all subsets of $X$) is a subset of the set of the fixed points of the generating function of the branching random walk. In particular here we are interested in the cardinality of the set of extinction probability vectors. We prove results which allow to understand whether the probability of extinction in a set $A$ is different from the one of extinction in another set $B$. In many cases there are only two possible extinction probability vectors and so far, in more complicated examples, only a finite number of distinct extinction probability vectors had been explicitly found. Whether a branching random walk could have an infinite number of distinct extinction probability vectors was not known. We apply our results to construct examples of branching random walks with uncountably many distinct extinction probability vectors.

In the long run, the eventual extinction of any biological population is an inevitable outcome. While extensive research has focused on the average time it takes for a population to go extinct under various circumstances, there has been limited exploration of the distributions of extinction times and the likelihood of significant fluctuations. Recently, Hathcock and Strogatz [PRL 128, 218301 (2022)] identified Gumbel statistics as a universal asymptotic distribution for extinction-prone dynamics in a stable environment. In this study, we aim to provide a comprehensive survey of this problem by examining a range of plausible scenarios, including extinction-prone, marginal (neutral), and stable dynamics. We consider the influence of demographic stochasticity, which arises from the inherent randomness of the birth-death process, as well as cases where stochasticity originates from the more pronounced effect of random environmental variations. Our work proposes several generic criteria that can be used for the classification of experimental and empirical systems, thereby enhancing our ability to discern the mechanisms governing extinction dynamics. By employing these criteria, we can i

Isolated populations ultimately go extinct because of the intrinsic noise of elementary processes. In multi-population systems extinction of a population may occur via more than one route. We investigate this generic situation in a simple predator-prey (or infected-susceptible) model. The predator and prey populations may coexist for a long time but ultimately both go extinct. In the first extinction route the predators go extinct first, whereas the prey thrive for a long time and then also go extinct. In the second route the prey go extinct first causing a rapid extinction of the predators. Assuming large sub-population sizes in the coexistence state, we compare the probabilities of each of the two extinction routes and predict the most likely path of the sub-populations to extinction. We also suggest an effective three-state master equation for the probabilities to observe the coexistence state, the predator-free state and the empty state.

This paper addresses the issue of how best to correct astronomical data for the wavelength-dependent effects of Galactic interstellar extinction. The main general features of extinction from the IR through the UV are reviewed, along with the nature of observed spatial variations. The enormous range of extinction properties found in the Galaxy, particularly in the UV spectral region, is illustrated. Fortunately, there are some tight constraints on the wavelength dependence of extinction and some general correlations between extinction curve shape and interstellar environment. These relationships provide some guidance for correcting data for the effects of extinction. Several strategies for dereddening are discussed along with estimates of the uncertainties inherent in each method. In the Appendix, a new derivation of the wavelength dependence of an average Galactic extinction curve from the IR through the UV is presented, along with a new estimate of how this extinction law varies with the parameter R = A(V)/E(B-V). These curves represent the true monochromatic wavelength dependence of extinction and, as such, are suitable for dereddening IR--UV spectrophotometric data of any resolu

We consider a system of two stochastic differential equations (SDEs) with competing two-way interactions driven by Brownian motions and spectrally positive $α$-stable random measures. Such a SDE system can be identified as a Lotka-Volterra type population model. We find nearly sharp conditions for one of the population to become extinct or extinguished.

We present a new method for deriving UV-through-IR extinction curves, based on the use of stellar atmosphere models to provide estimates of the intrinsic (i.e., unreddened) stellar spectral energy distributions (SEDs), rather than unreddened (or lightly reddened) standard stars. We show that this ``extinction-without-standards'' technique greatly increases the accuracy of the derived extinction curves and allows realistic estimations of the uncertainties. An additional benefit of the technique is that it simultaneously determines the fundamental properties of the reddened stars themselves, making the procedure valuable for both stellar and interstellar studies. We demonstrate how the extinction-without-standards curves make it possible to: 1) study the uniformity of extinction in localized spatial regions with unprecedented precision; 2) determine the relationships between different aspects of curve morphology; 3) produce high quality extinction curves from low color excess sightlines; and 4) derive reliable extinction curves for mid-late B stars, thereby increasing spatial coverage and allowing the study of extinction in open clusters and associations dominated by such stars. The

It is well-known that conditioning a supercritical (multi-type) branching process on the event that it eventually becomes extinct yields a subcritical branching process. We study the corresponding inverse problem: given a subcritical branching process, does there exist a supercritical branching process with the property that when we condition it on extinction, we get back the original subcritical branching process? We show that such a supercritical branching process (which we call a conjugate branching process) exists under mild hypotheses on the original subcritical branching process. We also show by example that if there are at least two types, then the conjugate branching process is not necessarily unique. Our results are relevant to the problem of constructing natural random planar maps whose scaling limit is given by supercritical Liouville quantum gravity. Moreover, conjugate branching processes can also be used to give alternative evolutionary hypotheses in cancer modeling.

This paper deals with an impulsive degenerate logistic model, where pulses are introduced for modeling interventions or disturbances, and degenerate logistic term may describe refugees or protections zones for the species. Firstly, the principal eigenvalue depending on impulse rate, which is regarded as a threshold value, is introduced and characterized. Secondly, the asymptotic behavior of the population is fully investigated and sufficient conditions for the species to be extinct, persist or grow unlimitedly are given. Our results extend those of well-understood logistic and Malthusian models. Finally, numerical simulations emphanzise our theoretical results highlighting that medium impulse rate is more favorable for species to persist, small rate results in extinction and large rate leads the species to an unlimited growth.

Extinction appears ubiquitously in many fields, including chemical reactions, population biology, evolution, and epidemiology. Even though extinction as a random process is a rare event, its occurrence is observed in large finite populations. Extinction occurs when fluctuations due to random transitions act as an effective force which drives one or more components or species to vanish. Although there are many random paths to an extinct state, there is an optimal path that maximizes the probability to extinction. In this article, we show that the optimal path is associated with the dynamical systems idea of having maximum sensitive dependence to initial conditions. Using the equivalence between the sensitive dependence and the path to extinction, we show that the dynamical systems picture of extinction evolves naturally toward the optimal path in several stochastic models of epidemics.

We consider continuous-state branching processes (CB processes) which become extinct almost surely. First, we tackle the problem of describing the stationary measures on $(0,+\infty)$ for such CB processes. We give a representation of the stationary measure in terms of scale functions of related Lévy processes. Then we prove that the stationary measure can be obtained from the vague limit of the potential measure, and, in the critical case, can also be obtained from the vague limit of a normalized transition probability. Next, we prove some limit theorems for the CB process conditioned on extinction in a near future and on extinction at a fixed time. We obtain non-degenerate limit distributions which are of the size-biased type of the stationary measure in the critical case and of the Yaglom's distribution in the subcritical case. Finally we explore some further properties of the limit distributions.

We provide a complete characterization of optimal extinction in a two-sector model of economic growth through three results, surprising in both their simplicity and intricacy. (i) When the discount factor is below a threshold identified by the well-known $δ$-normality condition for the existence of a stationary optimal stock, the economy's capital becomes extinct in the long run. (ii) This extinction may be staggered if and only if the investment-good sector is capital intensive. (iii) We uncover a sequence of thresholds of the discount factor, identified by a family of rational functions, that represent bifurcations for optimal postponements on the path to extinction. We also report various special cases of the model having to do with unsustainable technologies and equal capital intensities that showcase long-term optimal growth, all of topical interest and all neglected in the antecedent literature.

Understanding how an extinction event affects ecosystem is fundamental to biodiversity conservation. For this reason, food web response to species loss has been investigated in several ways in the last years. Several studies focused on secondary extinction due to biodiversity loss in a bottom-up perspective using in-silico extinction experiments in which a single species is removed at each step and the number of secondary extinctions is recorded. In these binary simulations a species goes secondarily extinct if it loses all its resource species, that is, when the energy intake is zero. This pure topological statement represents the best case scenario. In fact a consumer species could go extinct losing a certain fraction of the energy intake and the response of quantitative food webs to node loss could be very different with respect to simple binary predictions. The goal of this paper is to analyze how patterns of secondary extinctions change when higher species sensitivity are included in the analyses. In particular, we explored how food web secondary extinction, triggered by the removal of most connected nodes, varies as a function of the energy intake threshold assumed as the min

The Fisher-Stefan model involves solving the Fisher-KPP equation on a domain whose boundary evolves according to a Stefan-like condition. The Fisher-Stefan model alleviates two practical limitations of the standard Fisher-KPP model when applied to biological invasion. First, unlike the Fisher-KPP equation, solutions to the Fisher-Stefan model have compact support, enabling one to define the interface between occupied and unoccupied regions unambiguously. Second, the Fisher-Stefan model admits solutions for which the population becomes extinct, which is not possible in the Fisher-KPP equation. Previous research showed that population survival or extinction in the Fisher-Stefan model depends on a critical length in one-dimensional Cartesian or radially-symmetric geometry. However, the survival and extinction behaviour for general two-dimensional regions remains unexplored. We combine analysis and level-set numerical simulations of the Fisher-Stefan model to investigate the survival-extinction conditions for rectangular-shaped initial conditions. We show that it is insufficient to generalise the critical length conditions to critical area in two-dimensions. Instead, knowledge of the r

Established populations often exhibit oscillations in their sizes. If a population is isolated, intrinsic stochasticity of elemental processes can ultimately bring it to extinction. Here we study extinction of oscillating populations in a stochastic version of the Rosenzweig-MacArthur predator-prey model. To this end we extend a WKB approximation (after Wentzel, Kramers and Brillouin) of solving the master equation to the case of extinction from a limit cycle in the space of population sizes. We evaluate the extinction rates and find the most probable paths to extinction by applying Floquet theory to the dynamics of an effective WKB Hamiltonian. We show that the entropic barriers to extinction change in a non-analytic way as the system passes through the Hopf bifurcation. We also study the subleading pre-exponential factors of the WKB approximation.

We present a new model for extinction in which species evolve in bursts or `avalanches', during which they become on average more susceptible to environmental stresses such as harsh climates and so are more easily rendered extinct. Results of simulations and analytic calculations using our model show a power-law distribution of extinction sizes which is in reasonable agreement with fossil data. e also see a number of features qualitatively similar to those seen in the fossil record. For example, we see frequent smaller extinctions in the wake of a large mass extinction, which arise because there is reduced competition for resources in the aftermath of a large extinction event, so that species which would not normally be able to compete can get a foothold, but only until the next cold winter or bad attack of the flu comes along to wipe them out.