Analysis of a Length-Structured Model for Fish

Geigh Zollicoffer Author

08/05/2020 Added

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A mathematical study of the invasive white perch fish using a length-structured model.

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[00:00:03.002]Hello my name is Geigh Zollicoffer,
[00:00:04.788]And my UCARE Project was
[00:00:06.251]Analysis of a length-Stuctured
[00:00:07.746]Model for fish
[00:00:10.343]I was supervised by
[00:00:11.425]Dr. Richard Rebarber
[00:00:13.810]As some of you who fish may know
[00:00:15.545]the White Perch fish are not
[00:00:16.638]native to Nebraska
[00:00:18.413]and are actually also invasive
[00:00:21.341]The population accomplishes this by
[00:00:22.681]exhibiting stunting, or by dominating the
[00:00:25.219]overall population despite being the
[00:00:27.753]smaller fish
[00:00:29.601]In order to measure this
[00:00:30.945]we can observe the convergence
[00:00:32.045]of the total population,
[00:00:33.633]using a length based model.
[00:00:37.054]Our length based model that
[00:00:38.424]we used was proposed because
[00:00:39.869]it was:
[00:00:40.757]Easier to collect length data
[00:00:42.339]than age data.
[00:00:43.692]We thought that most anglers would
[00:00:45.337]be more interested in the size
[00:00:46.531]distribution rather than the age
[00:00:49.336]We also decided that length
[00:00:50.715]would be more applicable due to
[00:00:51.986]the life history parameters being
[00:00:53.641]more relatable
[00:00:55.100]to a length based model
[00:00:56.666]rather than an age based model
[00:00:59.562]Lastly, it was more tractable
[00:01:01.779]to mathematical analysis.
[00:01:06.203]One of the core elements of the length
[00:01:07.921]structured model is the
[00:01:09.298]population vector.
[00:01:11.567]The population vector shows the state
[00:01:13.370]of the population
[00:01:14.802]The vector uses a time
[00:01:16.335]step, which is a constant
[00:01:18.070]and is determined by the behavior
[00:01:19.609]of species or by data collection
[00:01:22.939]Each element is determined by length
[00:01:25.540]so P_0 would be new born fish
[00:01:27.521]and P_n would be the population
[00:01:30.186]of the largest fish possible
[00:01:32.661]Each time step a fish can either
[00:01:34.046]stay in it’s length class or
[00:01:36.058]grow into the next length class.
[00:01:41.024]In order to calculate P_t at a
[00:01:43.053]given time step, there
[00:01:44.916]are a few more parameters
[00:01:46.085]that we use.
[00:01:48.122]L_i which is the average
[00:01:49.578]length of fish at a specific ith stage
[00:01:52.283]S_i which is the survivability
[00:01:53.778]of a fish at a specific ith stage
[00:01:56.261]F_i which is the fecundity
[00:01:57.924]of a fish at a specific ith stage
[00:02:00.705]And lastly, p_t which is the
[00:02:03.204]probability that if ith fish survives
[00:02:05.840]it grows into the next stage
[00:02:10.948]Using these parameters, we
[00:02:12.296]were are able to use a model of the
[00:02:14.169]form P_(t+1), Which
[00:02:16.148]helps us traverse through
[00:02:17.492]time steps.
[00:02:18.714]As you can see the f_i’s are
[00:02:20.300]stored at the top
[00:02:21.571]and the diagonals are calculated using
[00:02:23.231]the determined survival rates
[00:02:24.563]and probabilities.
[00:02:26.881]This is similar to an age based
[00:02:28.677]model, however it is
[00:02:29.822]based on the length
[00:02:30.880]of the fish
[00:02:32.980]Concerning bio mass of a fish,
[00:02:34.713]the mass of the fish was computed
[00:02:35.946]using the length classes, and
[00:02:37.414]a coefficient alpha
[00:02:39.847]Therefore the total population biomass
[00:02:41.393]at a time step t is
[00:02:43.169]the total sum of the computed
[00:02:44.449]mass of a fish in class i,
[00:02:46.558]which is multiplied by the
[00:02:48.598]total population of fish in the same class
[00:02:52.488]There is also a Nonlinear system
[00:02:54.130]in place that is strictly
[00:02:55.200]decreasing and is continuous
[00:02:56.582]in order to calculate the probability
[00:02:58.148]of a fish growing larger.
[00:03:01.970]Looking at the dynamical system,
[00:03:03.932]we use a function g(B(t)) to calculate
[00:03:07.266]the probability.
[00:03:12.988]Therefore our model creates
[00:03:14.200]a system that:
[00:03:15.404]if there is low biomass, the probability
[00:03:18.502]that a fish gets larger is
[00:03:19.121]close to one, and concurrently
[00:03:20.721]if there is high biomass, the probability
[00:03:23.620]that a fish gets large in a time step
[00:03:26.204]is close to zero.
[00:03:28.016]These two implementations, help
[00:03:29.671]the model penalize crowding.
[00:03:33.753]This can be better shown by
[00:03:34.770]observing the nonlinearity of the system,
[00:03:38.226]where we are using a Beverton-Holt
[00:03:40.049]response function which gives
[00:03:41.525]the expected number of density
[00:03:42.859]in timestep t + 1 as a function
[00:03:45.294]of the density in the
[00:03:47.075]previous generation.
[00:03:49.752]We determine convergence in this
[00:03:51.647]model if vector P is
[00:03:53.314]globally asymptotically attracting
[00:03:55.215]for every nonzero, nonnegative P(0)
[00:03:59.803]Also known as the spectral radius
[00:04:01.488]we also use the largest eigenvalue of A
[00:04:04.253]which is denoted as p(A)
[00:04:06.351]to determine the asymptotic
[00:04:08.578]behavior of solutions after
[00:04:10.734]a certain number of time steps.
[00:04:16.488]Assume S_i and F_i are
[00:04:18.467]nondecreasing,
[00:04:19.452]Using p(A) and previous studies
[00:04:21.254]we can determine two
[00:04:22.355]outcomes of the population:
[00:04:24.396]if the zero population vector is
[00:04:26.116]asymptotically attracting,
[00:04:27.602]and if the system has
[00:04:28.846]a unique non zero equilibrium.
[00:04:31.948]However, the condition of s_i
[00:04:33.537]is often not satisfied,
[00:04:36.436]so we study if this theorem
[00:04:37.692]is true despite failing the nondecreasing
[00:04:39.668]condition.
[00:04:45.251]We do this by altering the
[00:04:46.619]survival rates to a non
[00:04:47.669]decreasing state
[00:04:48.752]This allows us to take a
[00:04:49.701]different approach to the main
[00:04:51.077]theorem and allows
[00:04:52.034]us to explore the results
[00:04:57.902]Each simulation was done with
[00:04:59.894]a different initial population
[00:05:01.910]than before and the
[00:05:03.426]limiting population is shown
[00:05:04.612]below.
[00:05:05.519]The first 15 trials are
[00:05:06.845]shown; however all
[00:05:07.777]tests had the same results
[00:05:09.026]in each example:
[00:05:11.925]This first type of survival rates
[00:05:13.327]simulated were sequential
[00:05:14.515]patterns
[00:05:15.396]giving convergence to zero.
[00:05:22.541]The second type of of survival rates
[00:05:23.864]simulated were a monotone
[00:05:25.590]like non decreasing pattern
[00:05:27.381]which showed convergence, this
[00:05:29.885]example shows convergence
[00:05:32.216]to a unique limiting population
[00:05:38.098]The third type of survival rates simulated
[00:05:39.810]were steep non decreasing
[00:05:40.705]patterns, that dropped
[00:05:41.940]down significantly as length class
[00:05:43.489]increased, this example
[00:05:45.089]shows convergence to a
[00:05:46.487]unique limiting pattern
[00:05:53.023]The final type of survival rates simulated
[00:05:55.031]were gentle non decreasing
[00:05:56.148]patterns, that
[00:05:57.205]dropped down a small
[00:05:58.371]amount over multiple length
[00:05:59.820]classes. This example
[00:06:00.987]converged to zero.
[00:06:09.543]After a number of simulations
[00:06:10.803]starting from different
[00:06:11.437]initial populations,
[00:06:12.936]testing the following statements
[00:06:13.919]from previous studies appear
[00:06:15.753]to be true, despite
[00:06:17.022]not following the non decreasing
[00:06:18.337]condition.
[00:06:22.374]Future work
[00:06:23.317]Since smaller fish are less
[00:06:24.548]desirable than larger fish
[00:06:26.938]we would like to identify survivals (si)
[00:06:28.921]that lead to limiting populations
[00:06:30.487]which are not dominated by small fish
[00:06:33.387]This might lead to management
[00:06:35.071]recommendations in the form
[00:06:35.959]of fishing regulations.
[00:06:40.702]This presentation was possible due
[00:06:42.987]to funding from the UCARE program
[00:06:46.156]thank you for
[00:06:47.203]your time.

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Analysis of a Length-Structured Model for Fish

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