# Growth and stability of fish populations

#### Greater Copenhagen area

Please see PDF for full description.

Project Background.
Estimating fish demography in the sea by trawling is
a standard method of assessing the size and health of fish populations, and has been used for many decades now. The age of fish scales with its weight w, or equivalently, size. Therefore, measuring the size allows to assess the age of fish, and demographers measure size distributions ranging from juvenile to adult fish. Under idealized equilibrium conditions, the size distribution of fish is a power law distribution, see Fig. 1 a). Such equilibrium distributions are useful in
assessing the ’health’ of a fish population, that is, to detect depletion of fish, e.g., due to disease or overfishing. An important question is therefore: how quickly does the fish population recover after perturbing it due, e.g., overfishing?

Project description. A model for the size distribution can be formulated on the base of size intervals, [w, w + dw]. Similar to mass conservation in fluid
dynamics, one then may set up a balance equation, as shown in Figure 1 b):
the abundance of a size group is a balance between how many individuals grow into the size group, how many grow out of it, and how many are dying. To solve the evolution of a size-structured population we must factor in the growth rate, because it sets the speed by which individuals move from one size class to the next. In a continuous size spectrum this balance is formalized in the so-called McKendrick-von Foerster partial differential equation.

Project aim.
We aim at addressing the questions:
1. How stable are size distribution against perturbations (eigenvalue spectrum)?
2. How fast does a fish population restore to its equilibrium size distribution, N = N (w)?
3. How does the dynamics between bony fish (the most common type of fish) and elasmobranchs (sharks, rays, etc) differ? These two types of fish
distinguish themselves in the way they make offspring: bony fish make offspring that is 1 mg regardless of the adult size, while elasmobranchs
make offspring roughly 1/100 of the adult size. The resulting dynamics is therefore qualitatively very different.

To do this, the project includes the following tasks: i) Implement a numerical solver for Eq. (1), to attempt analytical solutions and discuss their possible
failure. ii) Investigate eigenvalue spectrum and determine stability of different fish size groups. iii) Classify impact of various perturbations (e.g., overfishing of some size class) to the fish population.

## Requirements

Skills in programming, analytic and numerical simulation, interest in nonlinear dynamics and in biological systems.

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Contact

DTU Compute

#### Name

Erik Andreas Martens

Lektor

eama@dtu.dk

Supervisor info

### MSc in Aquatic Science and Technology

#### Supervisor

Erik Andreas Martens

#### Co-supervisors

Ken Haste Andersen, Mads Peter Sørensen

30

MSc thesis

### MSc in Mathematical Modelling and Computation

#### Supervisor

Erik Andreas Martens

#### Co-supervisors

Ken Haste Andersen, Mads Peter Sørensen

30

MSc thesis

### MSc in Physics and Nanotechnology

#### Supervisor

Erik Andreas Martens

#### Co-supervisors

Ken Haste Andersen, Mads Peter Sørensen

30

MSc thesis

### Technical University of Denmark

For almost two centuries DTU, Technical University of Denmark, has been dedicated to fulfilling the vision of H.C. Ørsted – the father of electromagnetism – who founded the university in 1829 to develop and create value using the natural sciences and the technical sciences to benefit society.

Today, DTU is ranked as one of the foremost technical universities in Europe, continues to set new records in the number of publications, and persistently increases and develops our partnerships with industry, and assignments accomplished by DTU’s public sector consultancy.

### Find us here

Anker Engelunds Vej 1
Bygning 101A
2800 Kgs. Lyngby

Denmark

Tlf. (+45) 45 25 25 25

CVR-nr. 30 06 09 46

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