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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[0:03] Hello my name is Geigh Zollicoffer,
[0:04] And my UCARE Project was
[0:06] Analysis of a length-Stuctured
[0:07] Model for fish
[0:10] I was supervised by
[0:11] Dr. Richard Rebarber
[0:13] As some of you who fish may know
[0:15] the White Perch fish are not
[0:16] native to Nebraska
[0:18] and are actually also invasive
[0:21] The population accomplishes this by
[0:22] exhibiting stunting, or by dominating the
[0:25] overall population despite being the
[0:27] smaller fish
[0:29] In order to measure this
[0:30] we can observe the convergence
[0:32] of the total population,
[0:33] using a length based model.
[0:37] Our length based model that
[0:38] we used was proposed because
[0:39] it was:
[0:40] Easier to collect length data
[0:42] than age data.
[0:43] We thought that most anglers would
[0:45] be more interested in the size
[0:46] distribution rather than the age
[0:49] We also decided that length
[0:50] would be more applicable due to
[0:51] the life history parameters being
[0:53] more relatable
[0:55] to a length based model
[0:56] rather than an age based model
[0:59] Lastly, it was more tractable
[1:01] to mathematical analysis.
[1:06] One of the core elements of the length
[1:07] structured model is the
[1:09] population vector.
[1:11] The population vector shows the state
[1:13] of the population
[1:14] The vector uses a time
[1:16] step, which is a constant
[1:18] and is determined by the behavior
[1:19] of species or by data collection
[1:22] Each element is determined by length
[1:25] so P_0 would be new born fish
[1:27] and P_n would be the population
[1:30] of the largest fish possible
[1:32] Each time step a fish can either
[1:34] stay in it’s length class or
[1:36] grow into the next length class.
[1:41] In order to calculate P_t at a
[1:43] given time step, there
[1:44] are a few more parameters
[1:46] that we use.
[1:48] L_i which is the average
[1:49] length of fish at a specific ith stage
[1:52] S_i which is the survivability
[1:53] of a fish at a specific ith stage
[1:56] F_i which is the fecundity
[1:57] of a fish at a specific ith stage
[2:00] And lastly, p_t which is the
[2:03] probability that if ith fish survives
[2:05] it grows into the next stage
[2:10] Using these parameters, we
[2:12] were are able to use a model of the
[2:14] form P_(t+1), Which
[2:16] helps us traverse through
[2:17] time steps.
[2:18] As you can see the f_i’s are
[2:20] stored at the top
[2:21] and the diagonals are calculated using
[2:23] the determined survival rates
[2:24] and probabilities.
[2:26] This is similar to an age based
[2:28] model, however it is
[2:29] based on the length
[2:30] of the fish
[2:32] Concerning bio mass of a fish,
[2:34] the mass of the fish was computed
[2:35] using the length classes, and
[2:37] a coefficient alpha
[2:39] Therefore the total population biomass
[2:41] at a time step t is
[2:43] the total sum of the computed
[2:44] mass of a fish in class i,
[2:46] which is multiplied by the
[2:48] total population of fish in the same class
[2:52] There is also a Nonlinear system
[2:54] in place that is strictly
[2:55] decreasing and is continuous
[2:56] in order to calculate the probability
[2:58] of a fish growing larger.
[3:01] Looking at the dynamical system,
[3:03] we use a function g(B(t)) to calculate
[3:07] the probability.
[3:12] Therefore our model creates
[3:14] a system that:
[3:15] if there is low biomass, the probability
[3:18] that a fish gets larger is
[3:19] close to one, and concurrently
[3:20] if there is high biomass, the probability
[3:23] that a fish gets large in a time step
[3:26] is close to zero.
[3:28] These two implementations, help
[3:29] the model penalize crowding.
[3:33] This can be better shown by
[3:34] observing the nonlinearity of the system,
[3:38] where we are using a Beverton-Holt
[3:40] response function which gives
[3:41] the expected number of density
[3:42] in timestep t + 1 as a function
[3:45] of the density in the
[3:47] previous generation.
[3:49] We determine convergence in this
[3:51] model if vector P is
[3:53] globally asymptotically attracting
[3:55] for every nonzero, nonnegative P(0)
[3:59] Also known as the spectral radius
[4:01] we also use the largest eigenvalue of A
[4:04] which is denoted as p(A)
[4:06] to determine the asymptotic
[4:08] behavior of solutions after
[4:10] a certain number of time steps.
[4:16] Assume S_i and F_i are
[4:18] nondecreasing,
[4:19] Using p(A) and previous studies
[4:21] we can determine two
[4:22] outcomes of the population:
[4:24] if the zero population vector is
[4:26] asymptotically attracting,
[4:27] and if the system has
[4:28] a unique non zero equilibrium.
[4:31] However, the condition of s_i
[4:33] is often not satisfied,
[4:36] so we study if this theorem
[4:37] is true despite failing the nondecreasing
[4:39] condition.
[4:45] We do this by altering the
[4:46] survival rates to a non
[4:47] decreasing state
[4:48] This allows us to take a
[4:49] different approach to the main
[4:51] theorem and allows
[4:52] us to explore the results
[4:57] Each simulation was done with
[4:59] a different initial population
[5:01] than before and the
[5:03] limiting population is shown
[5:04] below.
[5:05] The first 15 trials are
[5:06] shown; however all
[5:07] tests had the same results
[5:09] in each example:
[5:11] This first type of survival rates
[5:13] simulated were sequential
[5:14] patterns
[5:15] giving convergence to zero.
[5:22] The second type of of survival rates
[5:23] simulated were a monotone
[5:25] like non decreasing pattern
[5:27] which showed convergence, this
[5:29] example shows convergence
[5:32] to a unique limiting population
[5:38] The third type of survival rates simulated
[5:39] were steep non decreasing
[5:40] patterns, that dropped
[5:41] down significantly as length class
[5:43] increased, this example
[5:45] shows convergence to a
[5:46] unique limiting pattern
[5:53] The final type of survival rates simulated
[5:55] were gentle non decreasing
[5:56] patterns, that
[5:57] dropped down a small
[5:58] amount over multiple length
[5:59] classes. This example
[6:00] converged to zero.
[6:09] After a number of simulations
[6:10] starting from different
[6:11] initial populations,
[6:12] testing the following statements
[6:13] from previous studies appear
[6:15] to be true, despite
[6:17] not following the non decreasing
[6:18] condition.
[6:22] Future work
[6:23] Since smaller fish are less
[6:24] desirable than larger fish
[6:26] we would like to identify survivals (si)
[6:28] that lead to limiting populations
[6:30] which are not dominated by small fish
[6:33] This might lead to management
[6:35] recommendations in the form
[6:35] of fishing regulations.
[6:40] This presentation was possible due
[6:42] to funding from the UCARE program
[6:46] thank you for
[6:47] your time.

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Analysis of a Length-Structured Model for Fish

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