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To increase the production of proteins in East Africa, aquaculture gained increased attention recently. In this paper, we study the interactions of a consumer-resource system with harvesting, in which African Catfish (Gl ar i as gar i epi nus) consume a food resource. The cannibalistic behavior of African Catfish is captured by using a four stage-structured system. The dynamics of food resource and African Catfish result in a system of ordinary differential equations called a stage-structured fish population model. Existence and stability of steady states are analyzed quantitatively. We have investigated eight different harvesting scenarios which account for yield of the fish stock. Results from the simulations revealed that harvesting large juveniles and small adults under equal harvesting rates gives the highest maximum sustainable yield compared to other harvesting scenarios. In contrast to non-cannibalistic models, we find an increase of the proportion of the adult individuals under harvesting.

In this paper, a fishery model for African catfish and Nile tilapia is formulated. This model is used to compare financial profit and biomass outtakes in a two-species system versus single species systems. We consider a stage-structured fish population model consisting of the aforementioned fish species together with two food resources. The model dynamics include cannibalism, predator-prey, feeding, reproduction, maturation, development, mortality, and harvesting. We prove consistency of the model in the sense that the solutions will stay bounded and nonnegative over time. Conditions for local stability of fish-free equilibrium point are established. The simulation results reveal asymptotically stable solutions with coexistence of African catfish, Nile tilapia, and two food resources. The major conclusion from our findings is that fisheries should culture both species to maximize the biomass outtake and financial profit.

Modelling population dynamics in ecological systems reveals properties that are difficult to find by empirical means, such as the probability that a population will go extinct when it is exposed to harvesting. In this article, we use an aquatic ecological system containing one fish species and an underlying resource as our model. In particular, we study a class of stage-structured population systems, in both the deterministic and the stochastic settings, including stochasticity in such a way such that we allow the underlying resource to have a random growth rate. In these models, we study how properties connected to the fish species depend on different harvesting rates. To investigate models in the stochastic setting, we use Monte Carlo simulations to capture several of the emergent properties of the population. These properties have previously been studied in the deterministic case. In the stochastic setting, we get estimates for the expected outcome of population properties in our model, but we also get measures of dispersion. There are properties that emerge when introducing randomness in the model that cannot be studied in the deterministic cases, such as the probability of extinction. In this paper, we develop a method to derive this property. We also construct a method to determine the recovery potential of a species by introducing it in a virgin environment.

A great deal of ecological theory is based on simple Lotka-Volterra-type of unstructured population models in the study of complex population dynamics and communities. The main reason is to obtain important information for predicting their future evolution. In these unstructured models, it is assumed that all individuals in the population are identical, with the same birth and death rates, and consume equally from shared resources in a homogeneous environment. In reality, these assumptions are not biologically true but still forms a basis for modeling population ecology. We apply this paradigm on the grazing system consisting of coupled ordinary differential equations describing the dynamics of forage resource and livestock population in a grassland ecosystem. We do this by investigating the dynamics of the individuals at different life-history stages of juvenile and adult livestock. The mathematical derivation of the model is carried out to show how the physiologically structured population model can be approximated using a three stage-structured population model. Thus the resulting system of ordinary differential equations can be solved to predict density-dependent properties of the population since it provides a somewhat close-to-reality description of the natural and traditional grazing system. This model therefore certainly contains the needed information in the modeling methodology and accommodates the necessary amount of biological details about the population.

Estimates of tree heartwood and sapwood profiles are important in the pulp industry and for dynamic vegetation models, in which they determine tree biomechanical stability and hydraulic conductivity. Several phenomenological models of stem profiles have been developed for this purpose, based on assumptions on how tree crown and foliage distributions change over time. Here, we derive estimates of tree profiles by synthesizing a simple pipe model theory of plant form with a recently developed theory of branch thinning that from simple assumptions quantifies discarded branches and leaves. This allows us to develop a new trunk model of tree profiles from breast height up to the top of the tree. We postulate that leaves that are currently on the tree are connected by sapwood pipes, while pipes that previously connected discarded leaves or branches form the heartwood. By assuming that a fixed fraction of all pipes remain on the trunk after a branching event, as the trunk is traversed from the root system to the tips, this allows us to quantify trunk heartwood and sapwood profiles. We test the trunk model performance on empirical data from five tree species across three continents. We find that the trunk model accurately describes heartwood and sapwood profiles of all tested tree species (calibration; R-2: 84-99%). Furthermore, once calibrated to a tree species, the trunk model predicts heartwood and sapwood profiles of conspecific trees in similar growing environments based only on the age and height of a tree (cross-validation/prediction; R-2: 68-98%). The fewer and often contrasting parameters needed for the trunk model make it a potentially useful complementary tool for biologists and foresters.

Modelling population dynamics in ecological systems reveals properties that are difficult to find by empirical means, such as the probability that a population will go extinct when it is exposed to harvesting. To study these properties, we use an aquatic ecological system containing one fish species and an underlying resource as our models. In particular, we study a class of stage-structured population systems with and without starvation. In these models, we study the resilience, the recovery potential, and the probability of extinction and show how these properties are affected by different harvesting rates, both in a deterministic and stochastic setting. In the stochastic setting, we develop methods for deriving estimates of these properties. We estimate the expected outcome of emergent population properties in our models, as well as measures of dispersion. In particular, two different approaches for estimating the probability of extinction are developed. We also construct a method to determine the recovery potential of a species that is introduced in a virgin environment.

The Escalator Boxcar Train (EBT) is a commonly used method for solving physiologically structured population models. The main goal of this paper is to overcome computational disadvantages of the EBT method. We prove convergence, for a general class of EBT models in which we modify the original EBT formulation, allowing merging of cohorts. We show that this modified EBT method induces a bounded number of cohorts, independent of the number of time steps. This in turn, improves the numerical algorithm from polynomial to linear time. An EBT simulation of the Daphnia model is used as an illustration of these findings.

The Escalator Boxcar Train (EBT) is a commonly used method for solving physiologically structured population models. The main goal of this paper is to overcome computational disadvantages of the EBT method. We prove convergence, for a general class of EBT models in which we modify the original EBT formulation, allowing merging of cohorts. We show that this modified EBT method induces a bounded number of cohorts, independent of the number of time steps. This in turn, improve the numerical algorithm from polynomial to linear time. An EBT simulation of the Daphnia model is used as an illustration of these findings.

Branch formation in trees has an inherent tendency toward exponential growth, but exponential growth in the number of branches cannot continue indefinitely. It has been suggested that trees balance this tendency toward expansion by also losing branches grown in previous growth cycles. Here, we present a model for branch formation and branch loss during ontogeny that builds on the phenomenological assumption of a branch carrying capacity. The model allows us to derive approximate analytical expressions for the number of tips on a branch, the distribution of growth modules within a branch, and the rate and size distribution of tree wood litter produced. Although limited availability of data makes empirical corroboration challenging, we show that our model can fit field observations of red maple (Acer rubrum) and note that the age distribution of discarded branches predicted by our model is qualitatively similar to an empirically observed distribution of dead and abscised branches of balsam poplar (Populus balsamifera). By showing how a simple phenomenological assumptionthat the number of branches a tree can maintain is limitedleads directly to predictions on branching structure and the rate and size distribution of branch loss, these results potentially enable more explicit modeling of woody tissues in ecosystems worldwide, with implications for the buildup of flammable fuel, nutrient cycling, and understanding of plant growth.

The Escalator Boxcar Train (EBT) is one of the most popular numerical

methods used to study the dynamics of physiologically structured population models.

The original EBT model accumulates an increasing system of ODEs to solve for each

time step. In this project, we propose a merging procedure to overcome computational

disadvantageous of the EBT method, the merging is done as an automatic feature.

In particular we apply the model including merging to a colony of Daphnia Pulex.