
MBW:A stagebased matrix population model of invasive lionfish with implications for controlFrom MathBioContentsIntroductionLionfish are a nonnative inhabitant of the Western North Atlantic Carribean, and the study of their growth in this region is critical for determining a way to control their population. Originally from the IndoPacific region The lionfish are competing with native reef fishes that are already heavily exploited and as a result there has been an increased interest in finding a method to control this invasion. The goal of this work will be to work with a model that describes these dynamics with the goal of finding things that humans can change in the environment in order to regulate the lionfish population. Recent studies have shown that to bring the growth of the lionfish to a halt 27% of the adult population will need to be removed every month. Clearly this is an unreasonable goal, and ideally further research will indicate that this number is much lower in reality, and more realistic goals can be set to keep the lionfish population from continuing to decimate Atlantic reefs. We will build off a model created and analyzed by James A. Morris of NOAA, and James A. Rice of NC State University in an attempt to gain increased realism and more accurate long term predictions. The Original ModelThe original model utilized a great deal of accurate information related to lionfish reproduction and survival, however several avenues of thought went unexplored. We will be looking into how the parameters that Morris and Rice held fixed might affect the growth rate of the lionfish, and if changing these values might give rise to possible solutions to the lionfish control problem. We will also attempt to make the model more realistic by introducing seasonality by allowing particular parameters to oscillate over time. We will then propose a means by which we could possibly control the lionfish population based on our findings. Morris and Rice use the following agestructured model to predict the lionfish population over time: The variable values, their descriptions, values and relative elasticities are summarized in the following table: We begin the analysis by determining how this model works, and how the values of the Leslie matrix are calculated. To the left is a table that describes how each of these elements are calculated, and a description of what each of the parameters in the equations represent. The elements of the matrix were calculated based on a set of biological factors surrounding the lionfish. These values, taken from previous studies and available literature, represent the basic life cycle of the genus. Of note is that the determining factor in creating a Leslie matrix with a step size of one month was the fact that the larval duration is the shortest stage duration at 30 days long : ResultsUsing a simple iterative calculation method, population projections were made based on Morris and Rice's model. These projections were found to be suitable for short time scales, but did not accurately represent the full system determining the species growth. The system showed continuing unbounded growth, and did not include any seasonal terms to represent the varying rates of lionfish reproduction. The growth shown was purely logarithmic, and on a ten year time scale the model was ultimately unrealistic. Initial conditions were chosen based on estimated densities of lionfish populations per hectare. The plot below shows the original model's prediction on a logarithmic scale: Clearly there are terms missing from this model, which can hopefully be found and analyzed in later sections. Updating the ModelWhile the simplicity of the model used by Rice and Morris is appealing it ignores several important factors that ought to be taken into account. The most obvious of which are the facts that the population cannot grow in an unbounded way, and that the fecundity of the fish is not necessarily going to be constant throughout the year. In an attempt to remedy this we introduce two alterations to the model in the hopes of more accurately describing the lionfish population dynamics. These updates will take the form of a logistic growth term, and a term that represents the seasonality of the lionfish fecundity.
Incorporating Seasonal FecundityIn a 2015 study carried out by Gardner et al. it was found that the gonadosomatic index (GSI) for lionfish varied seasonally. The GSI is a measure of the ratio of the gonads to the entire fish by mass. It was determined that the GSI is a valid indicator for reproductive capabilities of the fish, and therefore can be used to model the fecundity of the Pterois genus. The image to the right shows the GSI values over time as they relate to water temperature. It can be seen that there are consistent peaks relating to both the maxima and minima of temperatures. This allows a sinusoidal wave to be fit to the fecundity of the fish using a few simple parameters. The lionfish reproduce in batches of 2,000 to 42,000 every three to four days, with a median value occurring around 24,000. The GSI values showed the maximum was about three times higher than the minimum, and the minimum was about half the value of the median. The resulting sine wave to model fecundity then has an amplitude 12,000, a mean of 24,000, and a period length of 6 months with maxima occuring in March and September, yielding the equation: Adding in just this term had tremendous effects on the total outcome of the model. Beyond being readily apparent in producing a different structure to the model projections, this addition reduced the projected lionfish population by approximately three orders of magnitude at the end of ten years. The result for this addition alone is that the removal rate necessary to bring the growth at or below unity is dramatically decreased. Results of Adding Only Seasonal Fecundity With the sinusoidality term in place, the needed removal rate to bring the dominate eigenvalue below unity drops down to 22.21% every month. Although removing more than 1 out of every 5 fish is still an unrealistic goal, it shows that the model is moving towards more realistic control goals. The below plot shows the projected growth given the same system and the same initial conditions as above, with only the addition of sinusoidal fecundity, compared to the original growth projections (left). It should be noted that the original dip in the population is a result of choosing the initial population to be only adults. Introducing Logistic GrowthWhen we first sought to introduce density dependence into the population growth we were immediately confronted with the question of what type of competition we would be modeling. While it might seem logical that scatter competition would best model fish populations in reality contest competition is far more suited for fish and especially for lionfish. The reason that this type of competition is likely to accurately describe lionfish populations is that these fish are very competitive when it comes to resources. In fact recently it has been observed that the lionfish off the Florida Keys have been eating one another! This lead us to use a term based on the BevertonHolt competition model, which is shown below. Note R is the growth rate, M the carrying capacity and N the population.
The result of adding the logistic growth term depends entirely on what value of 'b' we are using. Since there is no data on this value for the lionfish we used values for fish of similar size [Osenberg]. The typical range for this value was 10e4 to 10e2. Below we have plotted the population dynamics for 'b' set to 10e4:
Results of Combined UpdatesWhen both of these updates are applied to the model a much more realistic growth factor is seen over the ten year period investigated. The likely seasonal reproduction and density dependence exhibited by the lionfish shows that the projected population is much more controllable under certain conditions and removal rates. These outcomes show that the next major step in controlling the lionfish populations will require a better approximation for the density dependence of this genus in particular, and a removal rate to curtail the population can then be determined. As a case study calculations were performed to determine what the critical density dependencies would be for 0%, 5%, 10%, and 15% removal rates in order to bring the growth rate to one in a timely fasion (i.e. before 10 years). As of this writing there is no definitive value for density dependence of lionfish, but recent research show that similar reef dwelling fish have density dependencies between 10^2 and 10^4, so these values are within appropriate ranges. Below are the critical density dependencies as they correspond to potential removal rates: These conditions were then applied to the model and the growth rate for each case was plotted below: As can be seen above, these outlooks are much more realistic for controlling the population of lionfish. One important implication for the projection of these models is that the growth decay is not instant. There is a significant delay in time that removal starts and the time that the growth begins to slow towards unity. This implies that some control methods may need to be applied in advance of the desired results. ConclusionAfter incorporating both the seasonal fecundity, and the logistic growth the outlook was much more optimistic than the original model predicted. While the new estimates for the removal rate necessary to drop the largest eigenvalue below one are dependent upon the density dependence of the fish, most sets of reasonable values will eventually lead to a population that plateaus. This alone gives reason to believe that the model by Rice and Morris was an over estimate. This idea is further supported by the fact that once seasonal fecundity was introduced the population at the end of any given sampling time was decreased. Both of these results lend credence to the idea that the authors estimate of needing to remove 27% of the adult lionfish population is potentially higher than increased realism may suggest.
References[1]Benkwitt, Cassandra E. "DensityDependent Growth in Invasive Lionfish (Pterois Volitans)." PLoS ONE 8.6 (2013): n. pag. Web. [2]Gardner, Patrick G., Thomas K. Frazer, Charles A. Jacoby, and Roy P. E. Yanong. "Reproductive Biology of Invasive Lionfish (Pterois Spp.)." Front. Mar. Sci. Frontiers in Marine Science 2 (2015): n. pag. Web. [3]Morris, James A., and James A. Rice. "A Stagebased Matrix Population Model of Invasive Lionfish with Implications for Control." Biol Invasions Biological Invasions 13.1 (2010): 712. Web [4]Osenberg, C. W., St. Mary, C. M., Schmitt, R. J., Holbrook, S. J., Chesson, P. and Byrne, B. (2002), Rethinking ecological inference: density dependence in reef fishes. Ecology Letters, 5: 715–721. doi:10.1046/j.14610248.2002.00377.x [5]Yoneda, Michio, Haruhiko Miura, Manabu Mitsuhashi, and Michiya Matsuyama. "Sexual Maturation, Annual Reproductive Cycle, and Spawning Periodicity of the Shore Scorpionfish, Scorpaenodes Littoralis." Environmental Biology of Fishes 58.3 (2000): 30719. Web. Project Categorization(a) Mathematics Used: Matrixmodels are used to determine speciesremoval rates for making dominant eigenvalue less than one. Matching of sine function with data is done, and longterm behavior is simulated by simple matrix operations on initial conditions. (b) Type of Model: A simple stagestructured Leslie matrix is modified to include effects of seasonal variation in fecundity, and densitydependent contest competition. Fecundity is modified by a sinusoidal function matching the maxima, minima and median of fecundity data. For densitydependence, it is assumed that the adults eat some juveniles in a contest competition. Like in the original paper, mortalitydependence of populations is taken to be exponential. (c) Biological System Studied: The system being studies is the invasive population of lionfish in the Western NorthAtlantic Carribean. To protect the native fish, a large percentage of the lionfish needs to be removed every year. An earlier estimate of this percentage was quite high, and was based on a simple, shortterm model. This article considers a more detailed model including fecundity and density effects, which leads to a much lower required fishingrate. It is also observed that the populationdecay response has a finite delay after start of fishremoval. This removal strategy can be further optimized through monthly or daily fishingrate control. Data on the densityeffects was unavailable at the time of writing this article, so the results may need some modification. Citation of PaperGreen et al (2014), "Linking removal targets to the ecological effects of invaders: a predictive model and field test" In this paper, a sizestructured model of lionfish predation in the Bahamian reefs was constructed. This model was tested on multiple reefs over 18 months, to get threshold reduction in lionfish population for native species recovery. Smaller size species were found to recover faster, with a lagged response of larger fish. Due to variation in species populations across various reefs, thresholds were also very different. An important result of this study is that only partial removal of the invading species is required to recover native populations. The method for local determination of this threshold is outlined. [Green, Stephanie J., et al. "Linking removal targets to the ecological effects of invaders: a predictive model and field test." Ecological Applications 24.6 (2014): 13111322.] See Also
For more on stagestructured models and corresponding sensitivity analysis, see the following:
