Biological Invasions

, Volume 16, Issue 4, pp 903–917 | Cite as

Could an Asian carp population establish in the Great Lakes from a small introduction?

Original Paper


Bighead (Hypothalmichthys nobilis) and Silver carp (Hypothalmichthys molitrix) have established populations in the Mississippi, Illinois, Missouri and Maumee rivers, and because of the hydrological connections, there is now a risk that these species may establish in the Great Lake basins. It has been suggested this risk is minimal because of the small number of fish that breach containment measures, and possible mating-finding difficulties as a consequence. Using literature data, we parameterize a stage- and river-structured population model and examine the probability of a small number of fish establishing a population in one of the Great Lakes. We find that for sexual maturity earlier than age 5, there can be a significant risk of establishment with a very small number of fish (<20) in a lake basin with 10 or fewer spawning rivers. If all fish locate spawning rivers, mating is quite probable for very few spawning rivers. The subdivision of a few spawning adults across a large number rivers does reduce the probability of successful mating, but once a threshold number of fish is reached (dependent on the number of spawning rivers and the probability of fish locating a river), then mating success is very likely. Environmental stochasticity that reduces spawning success and juvenile survival predictably reduces establishment probability, but if spawning rivers have environmental conditions that fluctuate out of phase, this impact is much reduced. As expected, the most hazardous containment breach scenario is if barriers are continually leaky, and a small number of fish are introduced into the lake basin each year. In contrast, a single introduction represents a lower risk of establishment. Overall, the model suggests that establishment is quite likely (>75 % probability) for a large number of scenarios involving a small number of founding individuals (<20 fish). We conclude that while propagule pressure does increase risk in this system, it is not the most important consideration. Instead, probable age at first maturity in a given Great Lake basin may be critical to determining risk.


Asian carps Mate encounter Invasive species 


The Silver carp (Hypothalmichthys molitrix) and Bighead carp (Hypothalmichthys nobilis, previously Aristichthys nobilis), jointly known as Asian carps, are native to eastern Asia. Both species were introduced to North America in the 1970s as aquaculture fishes (Kolar et al. 2005), but escaped from captivity into the Mississippi River basin soon after (Kelly et al. 2011). The species have established dense populations in the Mississippi, Illinois, Missouri and Maumee rivers (O’Connell et al. 2011), and may also pose a threat to the Great Lakes. The proximity of the Asian carps to several hydrologic connections to the Great Lakes, including the Chicago Ship and Sanitary canal (Michigan) (Moy et al. 2011), Swan (Superior) and Maumee rivers, and the Erie canal (Erie) (USACE 2010), prompted the formation of the Binational Risk Assessment on Asian Carps in the Great Lakes. The authors of the assessment concluded that, if they successfully establish, Asian carp pose a substantial risk to the Great Lakes (Cudmore et al. 2012). We investigate the probability of establishment of these species using a stage- and river-structured population model.

Efforts to prevent these Asian carps from entering the Great Lakes include the construction of electric barriers on the Chicago Ship and Sanitary canal, and heightened border inspection to prevent live fish from being transported into Canada for sale in ethnic markets. There is, however, evidence that fish have crossed these barriers. In addition to the adult Bighead carp caught in Lake Calumet (Lake Michigan) in 2010, three adult Bighead carps in good condition were caught in Lake Erie by commercial fishers from 2000 to 2003, (Morrison et al. 2004; Cudmore and Mandrak 2011). Positive eDNA results (from genetic material in water samples) are regularly found beyond the existing electric barriers from Lake Calumet, the North Shore Channel of Chicago River, Des Plains River and Lake Erie (Sandusky and Maumee Bays) (Jerde et al. 2011, 2013; Rasmussen et al. 2011; Ohio DNR 2012).

Given that containment can never be perfect, the question arises as to whether there is a significant risk that Asian carp could establish a growing population in the Great Lakes. Some managers have contended that the number of fish required for successful spawning would be on the order of several hundred animals, and thus establishment is unlikely except under the circumstances of gross containment breach (e.g., widespread flood that connects river basins to the Great Lakes). In addition, successful spawning alone need not necessarily lead to the establishment of a population. It has also been suggested that Asian carps in the Great Lakes would be food-limited (Cooke and Hill 2010). The life history of Asian carps may also lead to reduced probability of establishment. These species may require large, fast-flowing rivers for successful spawning (Mandrak et al. 2011), and there are only a small number of suitable rivers on each lake (Mandrak et al. 2011; Kocovsky et al 2012). Juveniles then migrate from these rivers to wetland areas for approximately one year, before emerging to open waters as subadults. Low flow conditions could therefore prevent spawning or reduce juvenile survivorship. In addition, while the species are fast-growing and reach very large sizes, time to sexual maturity may be lengthened in colder waters (Kolar et al. 2005; Naseka and Bogutskaya 2011).

Therefore, in the event of accidental releases to the Great Lakes involving small numbers of fish (<50), one may conclude that positive population growth is unlikely to result given the difficulty in finding mates, environmental stochasticity, and potentially slow rates of maturation. While the movement rates of Asian Carps make it highly probable that fish will locate spawning rivers (Currie et al. 2011), the effect of the other potentially limiting factors on probability of establishment is unknown.

We start from the position that food limitation will not be a serious obstacle for either species of carp. While Cooke and Hill (2010) indicated that food limitation was a possibility in some locations, they also demonstrated that carp will have positive growth in most high productivity sites in the system, and further, they may have overestimated energetic costs. It is clear that, at least in some locations, Asian carps could find enough food (Cudmore and Mandrak 2011). We evaluate the probability of establishment of Asian carp from low density releases using models that incorporate stage-structured life history, division of spawning among different rivers, environmental stochasticity in river conditions and various containment breach scenarios. We find that establishment is quite likely for a wide range of scenarios, but may be sensitive to the age of first reproduction.

Materials and methods

Stage- and river-structured model of Asian carp population growth

We model population growth of Asian carp species using a stage- and river-structured population model with an annual timestep and pre-breeding census (Fig. 1). Similar stage-structured models have been used previously to describe invasion risks of aquatic species (e.g., Vélez-Espino et al. 2010). In each year, adult fish in the lake distribute themselves among the various rivers and spawn with a given probability related to environmental conditions and mate-finding. Juvenile fish in each river system then mature to subadults in associated wetlands, and then move into the lake population. We considered a range of maturation rates, resulting in 1–4 subadult stages. We parameterized the model using literature data but, because there is considerable range in data that produced overlapping parameter estimates for Bighead and Silver carp, we did not develop separate models for these species.
Fig. 1

Schematic of the stage-structured population growth model of Asian carps. Adult fish (A), spawn over a number of suitable rivers, producing juveniles fish (J) in each of these R watersheds. Juveniles produce subadults (SA1) in the lake basin with some survivorship probability, and similarly subadults in age class 1 given rise to subadults in age class 2 (SA2), up to 1 year prior to the age at first reproduction (SAn), and subsequently, sexually mature adults

In a structured population model, the stage categories correspond to life history stages. We grouped eggs, larvae and juvenile fish into one class (J) that represents the portion of life history spent in the spawning river and associated wetlands. Individuals in this class either fail to hatch, die over the winter, or survive and become part of the first subadult class (SA1) at the beginning of the next growing season. Survival over the year was related to mean body size in each class. At the beginning of the first subadult stage, fish move from the protected wetlands to the larger lake during the spring. Fish that survive to the final subadult stage (SAn) enter the reproductively mature adult class (A). We subdivided the subadult class (e.g., \(SA1, SA2, \ldots, SAn\)), but used a single division for adults because growth at smaller sizes affects survival more profoundly than at larger sizes.

If we do not consider the subdivision of the spawning adults and juveniles among separate rivers or environmental stochasticity, the dynamics of a low density population of female fish in the spring immediately before spawning, can be predicted as:
$$\left[ \begin{array}{l} J(t+1)\\ SA1(t+1) \\ SA2(t+1)\\ \vdots \\ SAn(t+1)\\ A(t+1) \end{array} \right] = \left[\begin{array}{llllll} 0 & \cdots & 0 & 0& F/2*P_{J}\\ P_{SA1} & 0 & \ldots & 0 & 0& 0 \\ 0 & P_{SA2} & \ldots & 0 & 0&0 \\ \vdots& \vdots& \ldots& \vdots& \vdots& \vdots\\ 0 & 0 & \ldots & P_{SAn}& 0 &0 \\ 0 & 0 & \ldots& 0&P_{A} & P_{A} \end{array}\right] \times \left[ \begin{array}{l} J(t) \\ SA1(t) \\ SA2(t)\\ \vdots \\ SAn(t)\\ A(t) \end{array} \right],$$
where F is the mean number of eggs produced per female fish (divided by 2 since we assume a 1:1 sex ratio), PJ is the probability of an egg resulting in a live juvenile fish in the river-wetland complex, PSA1 is the probability of a juvenile fish surviving the transition from the wetland to the open lake to become a subadult of age 1, PSA2 is the probability of a subadult surviving its next year in the open lake to reach age 2 and so on, while PA is the probability of an adult fish surviving from 1 year to the next. Mean values for these population parameters were taken from the literature as described below.

We used this version of the model to generate projected population growth rates for a range of fecundities and ages at first reproduction (as given by the dominant eigenvalue of the matrix). In addition, we completed a sensitivity analysis to determine which life history parameters had the most influence on model projections. The dependence of a projection on the particular parameter values was determined by calculating sensitivity, S, of each matrix element, aij as: \(S_{ij}=\frac{\delta\lambda}{\delta a_{ij}}=\frac{v_iw_j}{\sum_{i=1}^{st}v_iw_i}\) where st is the number of stages in the matrix, wj is the jth element of the dominant right eigenvector, and vi is the ith element of the dominant left eigenvector, and λ is the dominant eigenvalue. We then estimated the sensitivity for each individual parameter that made up the transition matrix elements, rk using the chain rule (Caswell 2001). The effects of proportional changes in the parameter values, or the elasticity, E, was then determined by the equation: \(E_{r_{k}}=S_{r_{k}}\frac{r_k}{\lambda}\). This calculation, and all others were completed in MATLAB (2011).

Model parameterization


Carp have fast growth rates and achieve large size (Jennings 1988; Johal et al. 2001; Schrank and Guy 2002). Three-year-old Bighead carp collected from the lower Missouri River in 1998–1999 averaged 550 mm in length, while 5-year-olds averaged 700 mm (Schrank and Guy 2002). Nuevo et al. (2004) found that 3-year-old Bighead carp in the Mississippi River ranged from 757 to 852 mm, and 5-year-old fish ranged from 807 to 909 mm.

Size estimates for younger ages seemed to be confined to lengths calculated from predicted growth relationships. Using a von Bertalanffy growth equation that predicts size at age, Nuevo et al. (2004) back-calculated average lengths as 273 mm at age 1, 500 mm at age 2, 672 mm at age 3, 830 mm at age 4, 813 mm at age 5, and 921 mm at age 6. Also in the middle Mississippi river, Williamson and Garvey (2005) back-calculated average lengths for Silver carp as 317.7 mm at age 1, 530.9 mm at age 2, 649.8 mm at age 3, 704.1 mm at age 4 and 723.3 mm at age 5. For Bighead carp at least, these sizes and growth rates are comparable to the native range, but higher than reported in other temperate areas (see references in Nuevo et al. 2004).

To parameterize the size of each stage in our structured population model, we also used a von Bertalanffy growth equation to predict size at age, \(L_t=L_{\inf}(1-e^{-K(t-t_0)})\), where Lt is length at time t, \(L_{\inf}\) is the asymptotic length at which the growth rate is zero, K is the growth rate, and t0 adjusts the equation for the initial size of the organism and is defined as age at which the organisms would have had zero size. Empirical data supports the use of this saturating growth curve. For example, Williamson and Garvey (2005) found that growth rates slowed as fish matured. The parameter values were based on all those that we found reported in the literature (Table 1).The majority of studies did not report errors about parameter estimates nor sample sizes, so we used the median of all reported values to parameterize the equation for the growth rate. We selected a value of 20 mm to represent the average length of the juvenile stage (age 0), and its correspondingly lower survival rate. In agreement with most reported values we used a single curve for males and females (e.g., Williamson and Garvey 2005).
Table 1

Reported parameters for the von Bertalanffy growth curve \([L_t=L_{\inf}(1-e^{-K(t-t_0)})]\) for Bighead and Silver carp











Amur river

Nikolskii (1961)




Gobindsager Reservoir

Tandon et al. (1993)




Middle Mississippi

Nuevo et al. (2004)




Middle Mississippi

Williamson and Garvey (2005)




Middle Mississippi

Garvey et al. (2007)





Middle Mississipp

Garvey et al. (2007)







There was little available data on survivorship, so we assumed that annual survivorship, PX, of the different different stages was related to body size. Survivorship was estimated using Lorenzen’s (1996) relationships between body weight, W, and natural mortality for juvenile fish, \(P_{J}=1-(1-e^{2.7*W^{-.315}})\), and \(P_{X}=1-(1-e^{3.30*W^{-.261}})\) for adult and subadult fish. The average weight values for each stage were calculated using published length–weight relationships (Table 2).
Table 2

Slopes and intercepts of reported length–weight relationships (log10(weight) = slope*log10(length) − intercept) for Bighead and Silver carp











Males, lower Missouri

Schrank and Guy (2002)




Females, lower Missouri




1998, middle Mississippi

Nuevo et al. (2004)




1999, middle Mississippi




1998, middle Mississippi channel only




Gavins point, Missouri

Wanner and Klumb (2009)




Interior Highlands, Missouri





Gavins point, Missouri




Interior Highlands, Missouri




Males, middle Mississippi

Williamson and Garvey (2005)




Females, middle Mississppi





Values for the slope and intercept of the relationship of log10(weight) = slopelog10(length) − intercept could vary with site, species and sex, but most cases we found reported very similar relationships. We used all relationships reported for North America (Table 2). In general, site effects seemed larger than species or sex effects. Wanner and Klumb (2009) report that length–weight relationships for both Bighead and Silver carp varied between the upper and lower Missouri. Williamson and Garvey (2005) give length–weight relationships for Silver carp that varied slightly between the sexes (Table 2). However, Schrank and Guy (2002) reported the relationship between weight and length was highly similar between male and female Bighead carp in the lower Missouri River (Table 2). For Bighead carp in the middle Mississippi, Nuevo et al. (2004) used two different relationships for the years 1999 and 1998, but found no differences between sexes.

Since the majority of studies did not report variance estimates about the regression coefficients, we simply used a single length–weight relationship for both sexes based on the median of the literature values of log10(weight) = 3.13 log10(length) − 5.35) to parameterize the model. The median values were quite similar to means obtained with weighting by sample size (slope = 3.03, intercept = −5.08), and moreover, were similar to values used in a different modelling exercise (slope = 3.27, intercept = −5.7 in Garvey et al. 2007).

Sexual maturity

Kolar et al. (2005) found that sexual maturity in Bighead carp is dependent on climate and may be reached between ages 2 and 7, with males often maturing one year earlier than females. In North American rivers (e.g., Schrank and Guy 2002), an average age of sexual maturity for Bighead carp may be age 2 or 3. Nuevo et al (2004) report mature males of age 2 in the middle Mississippi, and Williamson and Garvey (2005) also found that Silver carp in the middle Mississippi matured earlier (age 2) than in the species native range.

It is possible that colder waters, and lower food supplies, in the Great Lakes could reduce growth rates, and also increase age at first reproduction as compared to these river locations. The effects of delayed age at first reproduction could enter the model one of two ways: either separate from body size, or linked to growth rates and body size. In the latter case, if delayed sexual maturity is associated with slower growth rates, then in addition to potentially lower lifetime fecundity, lower average survivorship per individual would also be expected (given our assumption that survivorship is related to body size). Therefore, this scenario would produce lower population growth rates than the situation where reproduction was similarly delayed, but there was no corresponding decrease in growth rates.

We investigated this most conservative scenario where slower growth rates both delayed age of first reproduction and affected survivorship. We varied the growth parameter of the von Bertalanffy equation, and assumed that sexual maturity could not take place until individuals were greater than 400 mm (a conservative lower bound on size at 2 years from the North American literature). Therefore, variation in the growth parameter, K, produced variation in age at first reproduction ranging from age 2 to 5 (Table 3).
Table 3

Average size (mm) at stage for various age at first reproduction scenarios, calculated using the von Bertalanffy growth equation [Lt = 955(1 − eK(t−0.16))], where the value of K is indicated in parentheses


Age at first reproduction (K)

2 (0.4)

3 (0.3)

4 (0.2)

5 (0.15)

























We note that the median of the published parameter values suggested that carp would reach ∼550 cm by age 3, so that a population model with a juvenile stage, two subadult stages, and an adult stage (with an average size of 700 mm) may be most the appropriate, and after investigating the impact of varying age at first reproduction on the projected population growth rate, unless otherwise stated, we used this growth scenario for the model simulations.


The number of larval fish produced per mature female per year will be a function of the probability of spawning, and the number of eggs produced per female. Observations of the egg masses of both Silver and Bighead carp indicate that these species have very large potential reproductive output. In Europe and Asia, the number of eggs per female is reported to range from 145,000 to 4,329,600 (Alikunhi et al. 1963; Abdusamadov 1987; Singh 1989; Kamilov and Salikhov 1996; Kolar et al. 2005). In North America, Schrank and Guy (2002) report 226,213–769,964 eggs for Bighead carp in the Missouri river. Garvey et al. (2007) give a mean for Bighead carp (118,485 eggs) with a wide range of 4,792–1,938,333 eggs. In this same study, the authors report mean eggs for Silver carp at 269,388 eggs in a successful year (2004), with a range of 26,650–3,683,150. In the middle Mississippi river, Silver carp averaged 156,312 eggs per fish with a range of 57,283–328,538 eggs (Williamson and Garvey 2005).

Given the lack of data, we did not attempt to model egg production as a function of body size. Instead, we conservatively bracketed these reported ranges and used values of 15,000, 150,000, and 1,500,000 fertilized eggs as our lower, middle and upper estimates of mean fecundity. We assumed that sperm is not limiting for a mated pair. Unless otherwise stated, we used the middle value of 150,000 eggs for simulations.

Spatial subdivision of spawners and juveniles

Asian carps require rivers with particular characteristics (e.g., fast-flowing, >100 km in length) for spawning (Kolar et al. 2005). Evaluations suggest that there may be approximately 10 such rivers on each of the Great Lakes (Mandrak et al. 2011; Kocovsky et al 2012). Previous modelling work suggests that fish will have no difficulty in reaching such rivers in each basin (Currie et al. 2011). However, when population size is small, females may have difficulty in finding mates, since spawning partners could be divided over many rivers (e.g., 4 male and 4 female fish evenly divided over 10 rivers will result in no matings). The exact number of males and females in each river will depend on the number of spawning rivers and times, the population size for each sex and the joint probability distribution of the fish across the suitable spawning times/locations. We used our basic stage-structured model and incorporated subdivision of both spawners and juveniles across suitable rivers in order to examine the effect of this spatial effect on establishment probability. Our approach here is best described as spatially implicit, as compared to some spatially-explict stage-structured population models which describe the movement of a population across a continuous (e.g., Neubert and Caswell 2000) or patchy landscape (e.g., Sutherland et al. 2012)

Taking the simplest case of one spawning event per season, a 1:1 sex ratio and an even distribution across spawning sites, we calculated the expected number of mated females in a given river, Mi, as the integer value of the mean expectation approximated as \(E\langle{M_i}\rangle= (n-\sqrt{n(r-1)/\pi})({1}/{p_i})\) (see derivation and more general cases in Walter 1980), where there are r rivers, and a female fish has the same probability, pi, of being found in any of these spawning rivers (i.e., pi = 1/r for \(i= 1, 2, \ldots, r\)). The total number of male or female fish is n. So, for 50 male and 50 female fish distributed across 2 spawning rivers, summing over i, we expect 46 mated pairs, whereas for 20 possible spawning rivers, we expect only 32. We predicted the total number of spawning pairs, and where this number could not be divided evenly among the rivers, we used a random number generator to allocate the remaining number of mated pairs to rivers.

We note that this calculation assumes that all fish will be matched with a suitable river. That is, we assume that fish have no failure rate in locating a spawning river. If we instead assume that fish do not actively seek out rivers, and perhaps are only triggered to spawn if they happen to be close to a river when optimum conditions occur, or that the spawning cues are imperfect and can lead to the selection of an unsuitable river, then the number of mated pairs will be lower. In one approach, we can assume that there is given probability of encountering a spawning river during the correct environmental conditions (e.g., 20 % of fish locate a spawning river), and simply multiply the number of adult by this probability to obtain the number of potential spawners. However, a more interesting case results if the probability of locating a spawning river is a function of the number of rivers. We use a saturating function to describe the probability that an individual arrives at a spawning river. The function:
ranges from 0 to 1, where the probability of locating a river increases to the asymptote of 1 as s, the number of spawning rivers increases. The rate at which this probability increases depends on the value half-saturation constant, f. The value of this parameter gives the number of rivers at which the probability of an individual finding a river is 50 %. As f becomes large, a large number of spawning rivers are required for an individual to have a significant probability of locating one.

As a result of the subdivision of spawning pairs among rivers, the total number of juveniles produced in a given year would be the summation over the number produced in each river as: \(J{_i}(t+1)=\sum_{i=1}^{r}int\left(M_{i}(t)P_{J}\frac{F}{2}\right)\), where int indicates that we took the integer number of the predicted number of juveniles in each river-wetland complex (we also used the integer number of individuals in each spatially homogenous class), Mi is the number of mated females in a given river during the previous spawning period, and F and PJ are as previously defined.

We examined the effects of spatial subdivision by projecting forward from an initial condition where a number of adults had breached containment and entered a lake basin. We completed 10,000 simulations in order to calculate the probability that a population would establish. Unless otherwise stated, we define establishment as attaining a population of 1,000 females (an arbitrarily large value) in 20 years.

Environmental stochasticity

Spawning of Asian carp occurs in large, fast-flowing rivers (Jennings 1988; Costa-Pierce 1992; Kolar et al. 2005). In the early to late spring, as water level rises, fish move upstream to spawn (Verigin et al. 1978; Pflieger 1997). Conditions of high flow and warm water are most often correlated with spawning events in both species (Verigin et al. 1978; Pflieger 1997), and low flow conditions have been observed to correspond with year class failures (DeGrandchamp et al. 2007 for the Illinois river, and references therein). In the native range, in the Pearl river, from 2006 to 2008 the abundance of larval fish for both Bighead and Silver carp was significantly and positively related to discharge (Tan et al. 2010). However, in the Illinois river, DeGrandchamp et al. (2007) observed larvae over several months in a year where flow velocities only exceeded 0.7 m/s for 1 week, and these authors speculate that high river stage may augment larval survival, but that higher levels are not required for successful spawning. In agreement with this, both Lohmeyer and Garvey (2009) and Schrank et al. (2001) report that larval production coincided with quite small increases in discharge (about 100 m3/s) after water temperatures reached a suitable range (18–30 °C Nico et al. 2005; Jennings 1988; Schrank et al. 2001).

We assume that there are minimum conditions required for spawning which may not be met in a given year. However, because multiple spawning events per individual are possible (Pflieger 1997, Jennings 1988), and because the conditions required may actually be quite minimal, we also assume that spawning failure in a given year would be a relatively rare event for any of the Great Lake rivers. Therefore, we simulated environmental variability for each river using an autocorrelated random series of probabilities, generated using an AR(1) process and drawing values from a normal distribution, with a given variance and mean of 0.5. This sequence was subsequently truncated to lie between the values of 0 and 1 to serve as probabilities. Spawning only took place when the series value for that year, et, was greater than 0.05.

We also simultaneously examined the effect of environmental stochasticity on the two most sensitive population parameters, juvenile (J) and subadult stage 1 (SA1) survivorship. We assume that the same environmental factors that might affect spawning (e.g., low flow conditions) would affect juvenile survivorship and the survivorship of subadults at age 1, which are the most vulnerable when making the transition from the river-wetland system to the lake. At every timestep, the survival of juvenile fish and subadult fish of age 1 was related to the random probability describing environmental stochasticity in a given river by multiplying the survivorship PJ or PSA1 by et.

Large temporal variance has been implicated in species extinctions, and can have a similar negative impact on small, establishing populations. We therefore examined the effect of low environmental variance (σ = 0.2) and high environmental variance (σ = 0.8), and both correlated and uncorrelated variation between rivers, on the establishment probability (calculated from 1,000 replication simulations) over a range of initial population sizes and numbers of suitable rivers.

Containment breach scenarios

Since there are several possible pathways for fish introduction, we also considered alternate containment breach scenarios: a single release of a number of adults, or a single release of subadults of age 1, and a leaky barrier where either a number of adults or subadults of age 1 enter the lake each year. We evaluated establishment probability and time to establishment for combinations of these scenarios with environmental stochasticity, a limited number of spawning rivers, varying ability of fish to locate spawning rivers and varying age at first maturity using 1,000 replication simulations for each parameter combination.


Positive population growth rates, sensitivities and elasticities of the stage-structured model

Increased age at first reproduction can result in negative population growth rates when combined with a low mean number of eggs per female. In the absence of environmental variation and division of the juveniles among different rivers, negative population growth only occurred for the combination of a low mean number of eggs per female (15,000 eggs), and a slow growth rate which delayed reproduction until age 5 (Fig. 2).
Fig. 2

Projected annual population growth rate for various ages at first reproduction and fecundity, as given by the dominant eigenvalue of the related matrix, parameterized as described in the text (see Eq. 1)

The dependence of this projection on the particular parameter values was determined by calculating the sensitivity of each matrix element, the sensitivity for each individual parameter that made up the transition matrix elements, and the effects of proportional changes in the parameter values, or the elasticity (Caswell 2001). For all scenarios, population growth rate was most sensitive to juvenile survivorship, followed by survivorship as 1 year old subadults. Elasticity analysis suggests that the sensitivity to juvenile survivorship is primarily due to the small magnitude of this parameter value. When proportional changes are examined, adult survivorship has the largest impact on the population growth rate. However, since adult survivorship is already relatively large, it is likely that in natural populations, growth, and consequently probability of successful invasion, is controlled primarily by juvenile survivorship.

Effects of spatial subdivision between spawning rivers on establishment probability

In general, the subdivision of individuals across suitable rivers decreases the probability of finding a mate (Fig. 3a), but for reproduction at age 3 and fish that always locate rivers with suitable spawning conditions (f = 0), the establishment of a large breeding population is quite likely even for a relatively small number of introduced adult fish (Fig. 3b). As a result of difficulties in mate-finding associated with subdivision across rivers, establishment probability for a given initial number adults was lower for a large number of rivers.
Fig. 3

a The probability of finding a mate, calculated from 10,000 replicate simulations and an initial population of 20 females and 20 males. The probability of an individual locating a spawning river is either 100 % (half-saturation constant (f) equal to zero), or increases with the number of rivers until saturation at 100 %, where the speed with which this asymptote is approached depends on the magnitude of f. b The number of adult female fish required to have a 50 % probability of establishing a population of 1,000 adult females in 20 years, where individuals are sexually mature at age 3, as a function of the number of suitable spawning rivers

Where the probability of locating a suitable river increases with the number of rivers (f > 0), there is a peak in the probability of finding a mate at an intermediate number of rivers. The exact position of this peak will depend on the value of the half-saturation constant; however, we have no data to inform the value of this parameter. We assumed that carp would have a reasonable probability of locating a spawning river when more than two spawning rivers were present in a lake basin, and further, considered scenarios where more than 10 spawning rivers were required to be quite unlikely. As a result, a very large containment breach was required to have a 50 % establishment probability when there were fewer than 2 rivers (Fig. 3b). Where the number of suitable rivers was large (>10), for all scenarios the size of the required containment breach increased with the number of rivers.

Establishment probability with environmental stochasticity

When environmental variation is low, dividing reproductively active fish among many spawning rivers decreases the probability of establishment for a small initial number of adult fish. However, when a threshold number of introduced female fish is met (e.g., 10 fish for 10 spawning rivers), the probability of establishment increases abruptly to 100 %. For large environmental variance, the threshold number of fish required for positive establishment probability remains the same, but the increase in this probability with an increase in the initial number of fish is much slower (compare Fig. 4a, b).
Fig. 4

Probability of establishing a population of 1,000 female adult fish in 20 years, calculated from 1,000 replicate simulations, versus the number of adult female fish (assume equal number of males) that breach containment on one occasion for 1, 10 and 20 spawning rivers with environmental conditions that fluctuate independently between rivers for a low or b high environmental variance, and conditions that fluctuate synchronously among spawning rivers and c low or d high environmental variance

In addition, for environmental variance that is not correlated between rivers, establishment probability increases more quickly when there are more spawning rivers, after the threshold number of fish is exceeded (Fig. 4b). Presumably the multiple spawning locations buffer the population against the effects of bad years in a single river, which are more probable with large environmental variation. We can examine this effect by constraining all rivers to have exactly the same environmental variance. In this case, the increase in probability of establishment with an increase in the size of the containment breach is fastest for the smallest number of rivers (Fig. 4c, d).

Establishment probability for containment breach scenarios

We examined the effects of various types of containment breach for an intermediate risk condition of large environmental variance (σ = 0.8) that was uncorrelated among 10 spawning rivers, where all fish were able to locate a spawning river. We found that a continual annual leak of adults into a lake basin produces the highest risk of establishment for the lowest number of introduced animals (Fig. 5a). The scenarios where there was a one time breach of adults, or a continual annual leak of subadults at age 1 were equally risky, and the least risky scenario was a one time breach of subadult fish of age 1 (Fig. 5a).
Fig. 5

Establishment risk for various scenarios, where there are 10 spawning rivers that fluctuate asynchronously with large environmental variance (σ = 0.8), calculated from 1,000 replicate simulations for each parameter combination. a Probability of establishing a population of 1,000 female adult fish in 20 years versus either the number of female adult (A) or subadult fish age 1 (SA1), which have breached containment either on one occasion (annual), or every year (continual). We assume an equal number of males have breached containment. b Mean time in years to establish a population of 1,000 adult female fish (±1 SD) following a one time containment breach of a number of adult fish (assume equal number of males), where either all fish locate a spawning river (f = 0), or the number of fish that locate a spawning river is a saturating function of the number of rivers (f > 0; see Eq. 2). Only data from simulations where there was a 50 % or greater probability of establishment within 100 years are shown. c Mean time in years to establish a population of 1,000 adult female fish (±1 SD) where a given number of adult fish breach containment every year, all fish locate a spawning river, and fish are sexually mature at age 3, 4 or 5. Only data from simulations where there was a 50 % or greater probability of establishment within 100 years are shown

We compared establishment risk for the conditions where all fish locate a spawning river (f = 0) with the case where the number of fish that locate spawning rivers depends on the number of rivers (f > 0). In the scenario of large, uncorrelated variance among 10 spawning rivers and a one time breach of a number of adult fish, we find the shortest times to establish a population of 1,000 adult females for the case where all fish locate a spawning river (Fig. 5b). However, where river-finding is a saturating function of the number of rivers, establishment is very likely (50 % probability or greater) within 40 years for the range of saturation coefficients examined, where the number of introduced fish is relatively large (20 males and 20 females).

Finally, we examined the effect of age at first reproduction on establishment risk for large and uncorrelated variance among 10 spawning rivers, where all fish locate a spawning river, and there is a continual leak of adult fish. Mean time to establish a population of 1,000 adult female fish increased as age at first reproduction increased, although this time could be reduced with sufficient influx of fish (Fig. 5c). Establishment in 100 years was unlikely (<50 % probability) for fish that matured at age 5, when fewer than 100 fish (50 females and 50 males), breached containment annually. In comparison, establishment was likely within 20 years at all numbers of introduced fish when sexual maturity was reached at age 3.


We use a stage- and river-structured population model to demonstrate the risks of Bighead and Silver carp establishment in the Great Lakes following accidental introduction. We suggest that four factors may be of particular importance in increasing the risk of establishment: early sexual maturity, a small or intermediate number of spawning rivers, uncorrelated environmental stochasticity between rivers, and containment barriers which are continually leaky. Of these factors, the most influential may be early sexual maturity. We also note that for many of the scenarios examined, establishment probability is high for a relatively small number of introduced fish, therefore while propagule pressure plays a role in establishment risk, it is unlikely to have a large impact unless sexual maturity is delayed.

Our stage-structured population model, parameterized with literature data, suggests that positive population growth is the expectation for a range of parameter conditions, and that projected population growth is most sensitive to juvenile survivorship. While elasticity analysis suggested that proportional changes in adult survivorship would have the greatest impact on population growth rates, we note that adult survivorship is quite large in most populations, and therefore it is likely that the most sensitive stage to management or control actions is the juvenile stage. Relatively small decreases in juvenile survival will have relatively large effects on the viability and establishment of an Asian carp population, and in the case of control of an establishing population, this is probably the most promising target.

However, population growth rates are also very dependent of age at first reproduction, which is not easily manipulated in a stage-structured model, so we constructed a suite of stage-structured models with different ages of maturity (Eq. 1; Table 3). In the absence of food limitation or other site-specific limiting factors, our models parameterized with literature data suggest that for sexual maturity less than 5 years, our expectation should be for positive population growth of Asian carps in the Great Lakes. If carp have a prolonged and slow period of growth as subadults leading to age at first maturity greater than or equal to 5 years, then, for low fecundity, the deterministic growth rate may become negative, and population establishment is unlikely to occur. Correspondingly, when we examine the effect of age at first maturity on establishment probability where we incorporate the spatial segregation of fish among spawning rivers, fluctuations in environmental conditions that alter spawning probability, juvenile and subadult survivorship, and various containment breach scenarios, we also find that late sexual maturity substantially reduces the probability of establishment.

North American life history data that we were able to obtain is entirely from river systems (e.g., Schrank and Guy 2002; Nuevo et al. 2004; Williamson and Garvey 2005), that may have warmer waters and greater productivity which lead to fast maturation times that may not be characteristic of some Great Lakes populations. While very late ages at first reproduction (6+ years for females) have been reported for high northern areas of the Amur River system in Siberia where average air temperatures range from −7 to 0 °C (Gorbach and Krykhtin 1981; Naseka and Bogutskaya 2011), these cold extremes are not common in the Great Lakes. Mean annual air temperatures for this region range from 5 to 10 °C (NOAA,, and average water temperatures for the Great Lakes for 2004–2012 were: Superior 7.2 °C, Michigan 10.1 °C, Huron 9.4 °C, Erie 11.8 °C, and Ontario 10.7 °C. Silver carp likely require an average of 2,685 total annual degree-days to achieve maturity and Bighead carps, once mature, require an additional 655–933 degree-days for spawning (Gorbach and Krykhtin 1981; and summary in Naseka and Bogutskaya 2011). Both conditions are found in all of the Great Lakes except portions of offshore Lake Superior (Mandrak et al. 2011), though it should be noted that water temperatures are increasing, particularly in the spring season and the nearshore region (McCormick and Fahnenstiel 1999; Austin and Colman 2007; Trumpickas et al. 2008). Therefore we conclude that substantially delayed maturity is unlikely, but a determination of the effect of lake conditions on age at first maturity is of paramount importance in determining invasion risk for this species.

The relatively small number of suitable spawning rivers on each of the Great Lakes and the possible low probability of male and female fish simultaneously locating these rivers has been thought to be as a significant barrier to Asian carp establishment. Contrary to this intuition, a small number of possible sites to locate mates can be beneficial for establishment, since the aggregation of sexually active individuals at a small number of locations increases the probability of finding a mate. For Asian carp, we find that if all fish have the ability to locate suitable rivers when spawning conditions are good, populations can establish (defined as reaching a population size of 1,000 female fish in 20 years) with a small number of adult founding individuals when the number of rivers small, while a larger number of individuals is required to establish when there is a larger number of rivers (Fig. 4).

If, however, the probability that a fish is able to locate a suitable spawning river increases with the number of rivers, a large number of founding individuals is required to establish a population when there are a very small number of rivers on a given lake (Fig. 3b). Once the number of rivers has passed a threshold, which is related to the speed with which the river-finding probability approaches one, a larger number of rivers also requires a large number of introduced individuals for establishment. As a result establishment will be most likely in systems with a small to intermediate number of suitable spawning rivers.

Current evaluation of the number of suitable spawning rivers on the Great Lakes basins (Kocovsky et al. 2012; Mandrak et al. 2011) suggest that the numbers may be optimal for promoting the establishment of Asian carp (approximately 10 rivers on each lake). Moreover, carp may aggregate in productive sub-basin areas with one or two nearby spawning rivers, which may further elevate risk. Finally, these calculations assume no pheromone signalling system that would allow fish to detect the presence of the opposite sex at long distance. In the presence of such signals, mating success would be greatly elevated.

Large environmental stochasticity can decrease establishment probability by decreasing the geometric mean of population growth rates over a given period (Lewontin and Cohen 1969). We found that environmental stochasticity which altered spawning probability, juvenile and subadult survivorship did decrease establishment probability, but the effect was quite small for low variance. At larger variance, the magnitude of the decrease in establishment probability depended on whether fluctuations in river conditions were correlated or uncorrelated. Where variability was uncorrelated, establishment probability increased more quickly with the number of founding individuals when there was more than one river. An intermediate number of rivers, with conditions fluctuating out of phase represented the greatest risk, as the detrimental effects of bad environmental conditions on one river were ameliorated by good environmental conditions at another location. Presumably if there were a close relationship between spatial proximity and temporal correlation in environmental conditions, we should be able to assess risk on the different lake basins as a function of the spatial proximity of the various spawning rivers, with the greatest risk where rivers are widely spaced.

In conclusion, the models suggest that even a single event where a few adult individuals are accidentally introduced in one of the Great Lakes has, under some conditions, a significant probability of producing an established population of Asian carps. The number of adult fish in a single release required for close to 100 % establishment probability in 20 years depended on the exact scenario, but was usually less than 20 (10 males and 10 females). Establishment probability of a Great Lakes population was greater than 75 % for continually leaking containment where 10 adult fish were introduced every year (5 males and 5 females) and new individuals matured at age 3. Therefore, we suggest that establishment of Asian carps in the Great Lakes is extremely likely even for very small propagule pressure, unless age at first reproduction is substantially delayed.

Supplementary material


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • K. Cuddington
    • 1
  • W. J. S. Currie
    • 2
  • M. A. Koops
    • 2
  1. 1.Department of BiologyUniversity of WaterlooWaterlooCanada
  2. 2.Great Lakes Laboratory for Fisheries and Aquatic SciencesFisheries and Oceans CanadaBurlingtonCanada

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