https://orcid.org/0000-0003-4649-482X
Figures
Abstract
Invasive species threaten both ecological systems and economies. Successful establishment is governed partly by demographic stochasticity, where random individual-level variation can significantly influence invasion dynamics. A critical factor is the Allee effect, which describes reduced population growth at low densities, potentially serving as a barrier to persistence. Size-dependent growth and survival further shape outcomes based on the size distribution of individuals within the population. To address these complexities, we developed a population-level model integrating an Integral Projection Model (IPM) for the expected size distribution with an Allee effect for recruitment dynamics and a branching process to account for demographic stochasticity. We applied this framework to Silver Carp (Hypophthalmichthys molitrix), a species established in the Upper Mississippi River system that poses significant risk of expanding into the Great Lakes. Our analysis compared temporal dynamics and population-size distributions under hypothetical introduction scenarios for large adult and smaller subadult fish. In both cases, initial population size relative to the Allee threshold was a critical determinant of establishment probability, with introductions below the threshold requiring stochastic growth across a critical biomass boundary before self-sustaining dynamics could take hold. Sensitivity analysis further indicated that per-spawner recruitment played a compounding role in invasion risk: higher recruitment rates reduced the critical introduction size for establishment and compressed the time to self-sustaining population size, jointly broadening the window of invasion risk. These findings highlight the framework’s potential for invasion assessments and offer insights into how Allee effects, size structure, and recruitment rates interact to govern establishment outcomes, while underscoring the need for species-specific parameter estimates before quantitative outputs can guide management decisions.
Author summary
Invasive species can cause lasting ecological and economic damage, yet many introduced populations disappear before becoming established. Understanding why some succeed while others fail is therefore important for preventing future invasions. Small populations face particular challenges, since chance differences in survival and reproduction can determine their fate, and population growth is often difficult when too few individuals are present. We developed a model that tracks how the sizes of individuals in a population change over time while accounting for reduced growth at low population sizes and the role of chance. We applied it to silver carp, an invasive fish already established in parts of the Upper Mississippi River system and considered a potential threat to the Great Lakes. Across the scenarios we examined, the strongest predictor of establishment was whether the introduced population started above a critical population level. Populations that produced more offspring per spawner also established more easily and more quickly. Our approach may help assess invasion risk, though reliable predictions for any given species will require species-specific measurements before the results can guide management.
Citation: Peirce JP, Tam CS, Parkos JJ, Loppnow GL, Fritts MW, Coulter AA, et al. (2026) Combining branching processes and Allee effects into an integral projection model to assess invasion risk. PLOS Ecosyst 1(1): e0000022. https://doi.org/10.1371/journal.pesy.0000022
Editor: Frank E. Muller-Karger, University of South Florida, UNITED STATES OF AMERICA
Received: February 9, 2026; Accepted: June 27, 2026; Published: September 8, 2026
This is an open access article, free of all copyright, and may be freely reproduced, distributed, transmitted, modified, built upon, or otherwise used by anyone for any lawful purpose. The work is made available under the Creative Commons CC0 public domain dedication.
Data Availability: We present a new model. The only “data” are parameter values that come from previously published studies in the literature. We list these values in both the manuscript and the code that implement the model. The code supporting the model may be found at https://github.com/jppeirce/BranchingProcessIPM Furthermore, the model code runs independently without any data inputs and is self-contained.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Invasive species pose a significant threat to both the environment and the economy [1]. These non-native organisms, introduced either intentionally or accidentally into ecosystems, can disrupt local habitats, outcompete native species, and impose substantial ecological and economic expenses [2–4]. In order to successfully establish in a new environment, potential invaders have to overcome several challenges to avoid extinction. Small or low-density populations, in particular, are vulnerable to inverse density-dependent effects, which can arise from demographic stochasticity or limited cooperative interactions [5]. [6] first described how these factors can lead to a decline in the per-capita growth rate, a phenomenon now known as the Allee effect. In particular, an Allee effect can act as a natural barrier to establishment for some invasive species by reducing reproduction rates or increasing extinction probabilities at low population sizes [7,8]. However, species that overcome this initial hurdle often exhibit rapid population growth, leading to the potential for widespread ecological consequences [9]. While deterministic models often incorporate Allee effects by introducing an unstable equilibrium at low population levels, it is crucial to account for stochastic factors—random fluctuations inherent in natural populations—to fully understand extinction risks [10]. Models that incorporate demographic stochasticity are particularly effective in capturing the interaction between small population sizes and the Allee effect, especially during the early stages of an invasion.
Branching process models are probabilistic frameworks used to describe the growth and extinction dynamics of populations, where individuals reproduce independently according to specified probabilities [11]. In ecological and invasive species contexts, branching process models are employed to assess establishment probabilities under low-density conditions, incorporating factors such as the Allee effect and environmental stochasticity [12,13]. While there are many models that include the Allee effect for invasions and invasive species [14–16], few include demographic stochasticity. Some existing models combine branching processes with Allee effects, with applications including species invasion and lake system dynamics [17–20]. However, these approaches have not been extended to model establishment risk in a specific aquatic invasive species using size-structured population dynamics and a biomass-based Allee threshold.
In the model presented in this paper, we utilized a branching process to account for the stochastic nature of invasion when small population levels influence the probability of recruitment and survival. The dynamics of the expected size distribution is described by an Integral Projection Model (IPM). IPMs are particularly well-suited to studying species invasion because they bridge the gap between individual-level traits and population-level outcomes. In an IPM, future size states are predicted using size-dependent survival and growth functions of the current state, and the recruitment of new individuals from reproduction and immigration [21]. Moreover, IPMs allow for the incorporation of density-dependent processes, such as competition or the Allee effect, and can integrate environmental and demographic stochasticity. This enables users to assess their impact on both the probability of establishment and long-term population persistence which is an especially pressing concern for carp invasions in North America [22,23].
Silver Carp (Hypophthalmichthys molitrix) are native to eastern Asia and were introduced to the United States in the early 1970s to help control algae blooms in aquaculture ponds and sewage lagoons. Flooding events and accidental releases in the 1970s allowed the fish to escape into the Mississippi River and its tributaries, where they reproduced and established wild populations. Their invasion in the Upper Mississippi River region has significantly disrupted native fish populations, altered aquatic ecosystems, and posed economic challenges for commercial and recreational fisheries [24,25]. Cuddington et al. [22] investigated the probability of Silver Carp establishing in the Great Lakes following small-scale introductions using a stage- and river-structured individual-based model. Key findings indicated that establishment is highly probable (greater than 75%) in scenarios where fewer than 20 founding fish (10 males and 10 females) are introduced, particularly under conditions of early sexual maturity (before age five). The model highlighted that spatial subdivision of spawning rivers, environmental stochasticity, and the nature of containment breaches are critical factors influencing establishment risk.
In this study, we develop a size-structured stochastic modeling framework for species invasion that incorporates demographic stochasticity through a branching process and an Allee effect parameterized by population biomass, and apply this framework to Silver Carp establishment dynamics in the upper Mississippi River system. Our analysis examines the temporal population changes and size distributions of large adult and smaller subadult fish under hypothetical introduction scenarios. Through these scenarios, we evaluate the extent to which Allee effects may serve as a barrier to invasion, particularly for small founding populations, and identify key factors that influence the persistence of Silver Carp in novel environments. The scenarios were designed based on collective expertise, discussions among co-authors, and an understanding of conservation agency priorities.
Methods
Model
Set-up.
The model developed in this study formulates the expected length distribution of a female population as an IPM. In general, IPMs operate through discrete time steps, t, during which a census of individual sizes, z, in the population is recorded (for an introduction, refer to [21]). The state of the population at time t is represented by a continuous density function of the current size, n(z, t). Before the next census at t + 1, the size distribution undergoes an update, or projection, described by:
Here, is the interval of all possible sizes and
is an integral kernel constructed from size-dependent functions for survival, reproduction, and growth. The survival probability function s(z) can be modeled by logistic regression as survival data are commonly binomial with a dependence on size. Size also determines the probability that an individual is mature, represented by the function m(z). A growth function
maps from the current size, z, to the size at the next time step,
, using normal distributions with mean determined through linear regression of size and age-based data.
In this paper, we determine the reproduction and survival dynamics of mature females using a Ricker model with an Allee effect. The Ricker model is a widely used density-dependent framework for describing spawner–recruit relationships and remains a cornerstone of fisheries science (e.g., [26,27]). The Ricker population model with an Allee effect for a population size at time t has the following form [15]:
where is the intrinsic growth rate with carrying capacity K, and 0 < C < K is the Allee threshold. In this model, the zero equilibrium is asymptotically stable when the population size is small, below the threshold C. When initial values surpass C and remain below the maximum population size, the equilibrium K is asymptotically stable. Following standard practice in stock assessment and population modeling, we apply the Ricker formulation to spawning biomass rather than spawner counts. This choice reflects that fecundity often scales strongly with body size, making biomass a more appropriate predictor of reproductive output than simple abundance [28,29].
The time steps of a Ricker model are often generational, typically excluding the spawning stock from the next generation’s count. However, in an IPM, time steps are frequently smaller (e.g., days, months, years), allowing individuals to survive and be counted across multiple time steps. In addition, the biomass dynamics described by Eq 2 and the size-structured individual dynamics described by Eq 1 operate at different organizational scales, and connecting them requires a mechanism for translating population-level density dependence into per-individual reproduction. This mechanism is described below and is summarized by Fig 1.
Schematic of the three-phase projection from length distribution n(z,t) to . Phase 1: total biomass
is computed, the Ricker–Allee model Eq (2) predicts target biomass
, and surviving biomass
and potential recruit biomass
are obtained from the current population. Phase 2: the spawning proportion
is determined by the recruit deficit
relative to potential. Phase 3: each individual contributes to the next census through one of four mutually exclusive outcomes Eq (3), yielding the expected length distribution
via Eq (4).
Biomass and spawning
Given the current length distribution n(z,t) of the female population, the total biomass of the population at time t is
where W(z) is the weight of an individual of length z. Accounting for density-dependent effects, the Ricker–Allee model (Eq 2) predicts the target biomass at the next time step, . The target
generally differs from the biomass of individuals surviving to the next census, which depends on the size-dependent survival probability s(z). The surviving biomass at time t + 1 is
and the recruit deficit, , quantifies the additional recruitment required to attain the Ricker–Allee target. When the recruit deficit is negative, the surviving biomass alone exceeds the this target; consequently, reproductive output within the IPM framework is suppressed and set to zero. Conversely, for non-negative deficits, the level of adult spawning is determined by comparing the recruit deficit to the maximum potential biomass contribution from new recruits. The maximum biomass of new recruits,
, is defined as the product of the total number of mature adults,
, and the aggregate biomass of k female recruits per adult,
where denotes the length distribution of age-1 individuals, with
and
representing the mean and standard deviation, respectively. If the recruit deficit exceeds this maximum potential, such that
even universal spawning cannot bridge the gap. Consequently, all mature adults must reproduce to maximize recruitment output. When the recruit deficit is less than the maximum potential biomass (
), intermediate spawning is sufficient to achieve the target. The spawning proportion,
, is derived from the biomass balance equation:
. Thus, for the present year, the spawning proportion is given by:
The IPM developed here tracks the female size distribution only, with males assumed present in equal numbers throughout and in sufficient abundance to ensure fertilization success at all densities. Male availability is therefore treated as non-limiting, and the Allee threshold C reflects density-dependent reductions in population growth arising from mechanisms other than mate limitation.
Branching process
To compute the length distribution at the next time step, we partitioned the length interval into m subintervals of equal width
. In each subinterval (i.e., “length bin”) there are approximately a total of
individuals where
is a value sampled uniformly from
. The value of
can be separated into whole individuals and fractional individuals:
and
with
. A branching process is used to determine an individual’s contribution through survival and reproduction to the next census. In each length subinterval, we assume that the total number of individuals is
with probability
or
with probability
. This stochastic rounding is required only for discrete-individual simulations; the deterministic kernel derived below depends only on n(z,t) and is unaffected by the partition into whole and fractional individuals.
For each individual, we define the discrete random variable to indicate whether, during the current time-step, that individual dies without reproducing (X = 0), survives but does not reproduce (X = 1), reproduces but does not survive (X = k), or survives and reproduces (X = 1 + k). The probability of each of these events at the current time are length-dependent:
Expected length distribution
Based on the outcome of the random process, the expected contribution of a female with length in subinterval to the next length distribution is
Although each individual’s contribution is a random variable, individuals in subinterval i undergo independent branching outcomes, so by the law of large numbers the total contribution converges to its expectation as the population in each bin grows. Taking
refines the partition while preserving this convergence, supporting a deterministic equation for the expected length distribution at the next census. For all individuals in the subinterval, the total expected contributions are
And, for individuals of all lengths, as the subintervals become smaller (i.e., ) the expected length distribution at the next census,
Replacing the probabilities with their definitions, Eq 3, we finish with a model equation for expected length distribution,
The kernel of Eq 4 decomposes into two biologically interpretable terms. The first, , captures the survival and growth: individuals of length z survive with probability s(z) and transition to length
via the growth kernel
, regardless of reproductive status. The second,
, accounts for recruitment: an individual of length z is mature with probability m(z), a fraction
of the mature population spawns, and each spawning event contributes k recruits whose lengths follow the age-1 distribution
. The two-term kernel arises because survival and reproduction are assumed independent in the branching probabilities (Eq 3).
Numerical application: Silver Carp invasion
Silver Carp are planktivorous filter feeders native to freshwater systems in eastern Asia [30]. Their establishment in the Mississippi River Basin, including the Illinois River, followed escapes from aquaculture facilities in the 1970s [30,31], and their continued expansion poses a significant threat to the Great Lakes via the hydrological connection between the Illinois River and Lake Michigan [32]. Below, we describe the application of our model in this system.
Parameterization
To apply Eq 4 to an introduced Silver Carp population, the length-dependent survival, reproduction, and growth functions required parameterization. Silver Carp and related species have been the subject of substantial research owing to their invasion potential, providing a well-developed life history literature from which to draw. We adopted the IPM framework originally developed for Grass carp [33] and subsequently applied to Silver Carp [34,35], with parameter values sourced primarily from [36]. We assumed the same parameters values and incorporated the definitions for the growth kernel , the survival function s(z), and length distribution of new recruits,
provided in [35]. The probability that a fish of length z is mature was assumed to be described by the logistic sigmoid function,
, where the parameters are defined in [36]. The number of female recruits, k, depends on many factors such as the number of eggs produced (which is dependent in the size of the parent), the viability of those eggs, and the probability that an age-0 fish survives to be counted at the next sampling time. In the middle Mississippi River, Silver Carp exhibited an average fecundity of 156,312 eggs per fish with a recorded range of 57,283–328,538 eggs [37]. According to [35], the egg viability and survival probability of age-0 fish were estimated to be on the order of 10−5. Consequently, a single-digit value was selected for k in the simulations. We treated k as fixed to isolate demographic stochasticity from the branching process; allowing k to vary (e.g., Poisson or negative binomial) would mix reproductive and survival variation. Incorporating such variability to capture sweepstakes reproduction is a valuable direction for future work (Table 1).
Numerical experiments
Carp currently spread into new aquatic areas as adults, often through natural movement, but sometimes through intentional or unintentional human actions. For example, there is a documented instance of a Bighead Carp (Hypophthalmichthys nobilis) being purchased from a food market and released into local waterways as part of Buddhist mercy/life release rituals [40,41]. For large-scale release, contamination of fish purchased for stocking has led to the unintentional release of Bighead Carp into Chicago fishing lakes [42]. There is great concern that these various potential pathways may allow Silver Carp to disperse to the Great Lakes via connections to other basins, such as the Mississippi River basin [22].
Once introduced, abiotic factors such as water temperature, level, and flow may trigger spawning behavior in Silver Carp populations [43,44]. However, a key question exists as to the number of individuals required for a population to become established and for recruitment to occur. Unlike existing modeling approaches for Silver Carp [22,33], the IPM framework outlined above allows explicit representation of demographic stochasticity.
We conducted three numerical experiments to examine the effects of demographic stochasticity and Allee thresholds on Silver Carp establishment. Adult Silver Carp length was parameterized using the asymptotic length estimated for the Illinois River–Pool 26 confluence reach of the upper Mississippi River system ( = 0.867 m TL [37]), the primary invasion corridor through which Silver Carp disperse into the upper system. Age-1 subadult Silver Carp were estimated at 0.300 m TL, consistent with the average length at end of the first growing season reported for invasive populations across the Mississippi River basin [38]. At these lengths, the biomass of 10 adult fish is approximately 68 kg, equivalent in biomass to approximately 247 age-1 fish.
The Allee threshold C was examined at values corresponding to the biomass of 10, 50, and 100 adult fish as a sensitivity analysis spanning a range of founding event sizes considered plausible for accidental or incidental introductions. The lower bound is consistent with the introduction size at which [22] estimated approximately 50% establishment probability for Asian carp under favorable environmental conditions. The carrying capacity was set at the biomass 500 adult fish, chosen as a plausible and ecologically conservative upper bound relative to the Allee thresholds being considered in this study.
In the first experiment, we projected population dynamics forward from an initial condition at the Allee threshold biomass level of 10 adult fish using 1,000 replicate simulations over a 50-year horizon. This horizon was chosen to permit comparison with the asymptotic extinction probability results of [18], who demonstrate that extinction probability approaches 50% when the initial population is set at the Allee threshold. In the second and third experiments, we projected forward from initial conditions representing the arrival of Silver Carp into a lake basin across a range of Allee threshold values, estimating the probability of establishment and the time required for establishment, respectively, for introductions of adult and age-1 subadult fish, each estimated from 100 replicate simulations per introduction size. Establishment was defined as reaching a population of 1,000 adult females within 20 years, following the operational definition of [22]. The supplementary sensitivity analysis of both quantities with respect to k, the number of female recruits produced per spawning event, likewise used 100 replicate simulations per combination of introduction size and recruitment rate.
We numerically implemented the model with integral equations approximated by the Midpoint Rule, using Python (Version 3.10). Key packages include Scipy (Version 1.13.1), Numpy (Version 3.23.5), Plotnine (Version 0.14.0), and Pandas (Version 2.2.2). Model code is available at https://github.com/jppeirce [45].
Results
For the stochastic Ricker model with Allee effect,[18] demonstrated that the probability of extinction asymptotically approaches 50% when the initial population is set to the Allee threshold. Most simulations initialized at the threshold (C = 10 adult fish) maintained small population sizes for the first 10 years (Fig 2A). One sample path displayed large growth at year 15, whereas some simulations did not see large changes in size until after year 20. The probability of extinction after 50 years was . Simulated populations that did not become extinct by year t = 50 varied around the carrying capacity in a bell-shaped distribution (Fig 2B), consistent with the numerical examples in [18]: for initial biomasses close to C, there was a positive probability of extinction even when total population sizes exceeded the threshold.
A, C and E: Sample paths and the mean (+/- standard error) of the stochastic model for an initial biomass of female Silver Carp (10 adult, 247 subadult fish, 10 subadult fish) with an Allee biomass threshold of 10 adult 247 subadult fish. The stochastic mean is shown as a function of time (green dot-dashed curve), alongside five sample paths of the stochastic model (solid blue curves). The dotted red line represents the threshold. B, D, and F: Histograms approximating the frequency distribution of the stochastic model at t = 50 years, with bin widths of 200 fish.
When the initial distribution consisted of 247 age-1 fish of equal biomass, a strikingly different result was observed (Fig 2C and 2D). After a short transient period, age-1 fish matured to adults, increasing the likelihood of sweepstakes reproductive success. This boom-or-bust recruitment is known to occur in invasive carp [46]. Here , with most simulated population totals varying symmetrically around the carrying capacity after 20 years. When the initial number of age-1 fish was reduced to 10 individuals, the probability of extinction increased substantially to 0.784 (Fig 2E and 2F).
When investigating the dependence on the initial number of fish, a sigmoidal probability of rapid establishment resulted from an inflection point corresponding to the critical population size at the Allee threshold (Fig 3). When the threshold was small, a population biomass well above the threshold showed little to no Allee effect (demonstrated by a primarily concave-down graph). A more pronounced sigmoidal shape emerged when the initial population biomass was near the threshold. For many scenarios, the establishment probability was high for a relatively small number of introduced fish, suggesting that Allee effects were unlikely to substantially limit establishment unless the initial invasion was close to the biomass threshold for growth. Correspondingly, when the initial introduction was well above the threshold biomass, the establishment probability approached was high and the Allee effect again had little influence on outcomes. Results were similar when an equal biomass of age-1 fish was introduced.
A: Probability of establishing a population of 1,000 female fish within 20 years, calculated from 100 replicate simulations, as a function of the number of adult female fish breaching containment on one occasion, for Allee thresholds set at the biomass of 10, 50, and 100 adult fish, respectively. B: Mean time (1 standard deviation) in years to establish a population of 1,000 adult female fish following a one-time containment breach, as a function of the number of introduced adult females, for Allee thresholds set at the biomass of 10, 50, and 100 adult fish.
Across the three Allee threshold values, mean time to establish a population of 1,000 adult females decreased as the number of introduced females increased beyond the threshold biomass (Fig 3B). Mean time to establishment increased with the Allee threshold biomass, though larger introduction sizes reduced this effect. When the threshold was set at the biomass of 100 adult fish, establishment within 10 years was unlikely for the range of introduction sizes examined. When the threshold corresponded to the biomass of 50 or fewer adult fish and the initial introduction exceeded 75 females, establishment within 10 years was probable.
Per-spawner recruitment played a compounding role in invasion risk through two distinct mechanisms (Fig 4). The inflection point in establishment probability near the Allee threshold of 50 adult fish reflected a transition between two qualitatively different regimes: introductions below the threshold had to grow stochastically across the threshold before establishment could occur, while introductions above it proceeded under largely deterministic growth dynamics. Increasing the number of female recruits per spawning event reduced the critical introduction size at which establishment probability reaches approximately 50%, from roughly 50 females when a single recruit is produced per spawning event to approximately 40 females when 5 or 10 recruits are produced. This same threshold transition explained the maximum in mean time to establishment in Fig 4B, where the longest and most variable establishment times corresponded precisely to founding populations with biomass below the Allee threshold that nonetheless succeed, having spent an extended period in the stochastic near-threshold regime. Once introduction biomass exceeds C, mean time to establishment declined sharply, further compressed by higher recruitment rates. Taken together, these results indicated that even modest increases in per-spawner recruitment broadened the window of invasion risk by lowering the threshold introduction size and reducing the time available for detection and management response.
A: Probability of establishing a population of 1,000 female fish within 20 years, calculated from 100 replicate simulations, as a function of the number of female recruits. The number of female (k) recruits set as 1, 5, and 10 per spawning event, respectively. B: Mean time (1 standard deviation) in years to establish a population of 1,000 adult female fish following a one-time containment breach, as a function of the number of females recruits, for 1, 5, and 10 female recruits per spawning period.
Discussion
Integral Projection Models (IPMs) are well suited for studying invasion establishment as they link individual traits (e.g., length, age, reproductive condition) to population‐level processes such as survival and reproduction [21]. They also support evaluation of management actions by simulating how targeted removal, barriers, or habitat modifications propagate through these processes [35,47]. In this study, we integrate density-dependent processes, such as the Allee effect, into the IPM framework while using a branching process to account for stochastic reproduction inherent in low-density spawning situations. Our model explicitly accounts for size-dependent demographic pathways that are critical for predicting invasion risk.
Our simulations demonstrate that invasion risk for Silver Carp depends strongly on whether arrivals consist of adults or age-1 subadults. A small number of age-1 fish has a low probability of establishing a population (Fig 3A), whereas the introduction of a few adults or several hundred age-1 subadults of equivalent biomass substantially increases establishment risk (Fig 3). These findings provide managers with length-specific invasion thresholds that can inform surveillance and rapid-response strategies. For example, prioritizing measures that reduce adult dispersal or prevent large pulses of subadults could disproportionately lower establishment risk. In the Great Lakes context, ongoing efforts to maintain low densities near the electric barrier between the Illinois River and Lake Michigan [48] can be evaluated more precisely by translating length-specific propagule pressure and size composition into establishment probability using our framework.
Propagule pressure remains a central driver of establishment for aquatic invaders [49,50], including Silver Carp in the Great Lakes [22]. Movement models (e.g.,[51]) can provide system-specific estimates of propagule pressure and length composition, which our model can then map to establishment risk. However, translating these estimates to precise establishment probabilities requires an empirical estimate of the Allee threshold C for Silver Carp, which remains poorly constrained. As Fig 3 illustrates, establishment probability is sensitive to the magnitude of C, particularly for introduction sizes near the threshold, making empirical characterization of the Allee effect strength a critical data need for applied use of this framework. Such estimates could be derived from controlled introductions in mesocosms, field observations of founding populations, or retrospective analysis of documented invasion events. Coupled with tools that aim to reduce propagule pressure, such as targeted removals and engineering controls [52,53], a well-constrained estimate of C would allow these risk estimates to guide resource allocation and intervention timing with greater confidence.
The IPM developed here tracks the female size distribution only, with males assumed present in equal numbers and in sufficient abundance to ensure fertilization success at all densities. At very low founding densities, coordinated spawning failure and mate-finding limitation are recognized mechanisms of Allee effects in fishes and could act as additional barriers to establishment beyond those captured by C, suggesting the model may conservatively underestimate true establishment thresholds. Incorporating sex-structured dynamics and explicit mate-finding functions would be a natural extension of this framework.
Conclusion
While invasion or accidental introduction of small age-1 fish are still unlikely to include enough individuals for establishment, stochastic natural dispersal or intentional stocking of adults would likely increase the probability of successful recruitment. Relative to previous establishment models [13,18,54,55], our approach uniquely combines (i) length-based vital rates within an IPM, (ii) explicit Allee effects in a biomass-based Ricker formulation, and (iii) demographic stochasticity via a branching process tailored to low-density reproduction. This advances earlier Silver Carp risk assessments that focused on environmental stochasticity and counts of arriving females [22] by explicitly representing demographic stochasticity and size dependence. Although demographic heterogeneity among individuals is not yet included, this source of uncertainty can be important in small introductions [56] and represents a natural extension, particularly given evidence of life-history variation in Silver Carp across invaded systems [57,58].
Future work should focus on (i) embedding the model within a meta-population framework to estimate site-specific establishment risk across connected waterways [22], (ii) conducting uncertainty quantification and global sensitivity analyses to identify key parameters [59], (iii) validating the presence and strength of Allee effects in Silver Carp populations, and (iv) performing robustness checks comparing biomass- and number-based stock–recruit formulations. The scenarios analyzed in this study are hypothetical, designed to illustrate how length structure and the Allee threshold jointly shape establishment dynamics within our framework; translating these qualitative patterns into system-specific risk estimates will require empirical constraints on the parameters identified above. These extensions may enhance the applicability of our framework for decision-making and improve its predictive reliability.
Acknowledgments
Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government.
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