Double Trouble at High Density: Cross-Level Test of Resource-Related Adaptive Plasticity and Crowding-Related Fitness

Population size is often regulated by negative feedback between population density and individual fitness. At high population densities, animals run into double trouble: they might concurrently suffer from overexploitation of resources and also from negative interference among individuals regardless of resource availability, referred to as crowding. Animals are able to adapt to resource shortages by exhibiting a repertoire of life history and physiological plasticities. In addition to resource-related plasticity, crowding might lead to reduced fitness, with consequences for individual life history. We explored how different mechanisms behind resource-related plasticity and crowding-related fitness act independently or together, using the water flea Daphnia magna as a case study. For testing hypotheses related to mechanisms of plasticity and crowding stress across different biological levels, we used an individual-based population model that is based on dynamic energy budget theory. Each of the hypotheses, represented by a sub-model, is based on specific assumptions on how the uptake and allocation of energy are altered under conditions of resource shortage or crowding. For cross-level testing of different hypotheses, we explored how well the sub-models fit individual level data and also how well they predict population dynamics under different conditions of resource availability. Only operating resource-related and crowding-related hypotheses together enabled accurate model predictions of D. magna population dynamics and size structure. Whereas this study showed that various mechanisms might play a role in the negative feedback between population density and individual life history, it also indicated that different density levels might instigate the onset of the different mechanisms. This study provides an example of how the integration of dynamic energy budget theory and individual-based modelling can facilitate the exploration of mechanisms behind the regulation of population size. Such understanding is important for assessment, management and the conservation of populations and thereby biodiversity in ecosystems.

Observation. During the time-course of simulations, the total number of daphnids and the abundance within three size classes (small, medium, large) are observed. In the original experiments, size classes were obtained by sieving daphnids through mesh filters of various sizes. Preuss et al. [6] calculated the range of body length per size class based on the mesh diameter, multiplied by a length-to-width factor for daphnids of 1.6, assuming that daphnids passed through the mesh with their smallest dimensions. However, Martin et al. [7] found that a conversion factor of 1.6 overestimates actual body sizes. We therefore empirically determined size classes by sieving daphnids through the meshes used in the original study and subsequently measured daphnid body length excluding the spine, for the medium and large size classes (n = 100 each). We use the lower 95% confidence interval of measurements determined for medium and large size classes as lower class boundaries respectively, corresponding to size classes of: <1.25 mm (small), ≥1.25 -<2.1 mm (medium) and ≥2.1 mm (large). These values are close to the size classes used by Martin et al. [7].
To compare the composition of simulated populations and experimental results, we relate the actual body length measure L w to structural length by using a shape correction coefficient which translates physical length to volumetric length: For Monte-Carlo simulations, the mean and 95% confidence interval are calculated for total abundance and abundance within each size class based on values recorded each day for each simulation run.

Initialisation
Initialisation of model simulations is based on the experimental conditions they are supposed to represent, i.e. the volume of the environment (test vessel) as well as the abundance and the composition of the population at the start of the experiment. The simulations start with five neonate daphnids, less than one day old, and in addition, three adult daphnids, between two and three weeks old. To mimic these initial conditions, we simulate growth, maturation and reproduction as well as survival, until each daphnid reaches its respective age, starting with a body length at birth of 0.9 mm. For adults, a random age at simulation starts between 14 and 21 and is drawn from a uniform distribution. To represent food availability under culture conditions, we used a fixed scaled functional response of f = 0.7 (see submodel section), which fits growth measured under culture conditions well (unpublished result). Analogous to the experiments, adult daphnids are assumed to 5 be egg-carrying, one with new egg (embryo maturity level U H = 0), and two adults with furtherdeveloped eggs (embryo maturity level randomly drawn from a uniform distribution between 0 and 0.005).

Input data
The amount of food added to the system is included as input data to the model, representing the major external driving process in the laboratory setting. In accordance with the experimental scenario, food is added to the modelled environment daily on weekdays, with the amount tripled on fridays and no food is added during weekends. We used the two different scenarios of low food (0.5 mg total organic carbon (C) per day and population) and high food (1.3 mg C per day and population).

Submodels
Feeding and assimilation. The food ingestion rate is assumed to follow a functional response type II [8,9]: Where is the maximum ingestion rate for a filtration rate and is determined by the incipient limiting level [10]: Change in environmental food density X is due to the summed food ingestion of all daphnid individuals i: Ingested food is assimilated with a certain efficiency p x . The fraction of food that is not assimilated by the daphnid is excreted from the gut. In the model, we do not follow possible recycling of faeces and assume that these are immediately removed from the system. The assimilation rate is determined by the ingestion rate and the assimilation efficiency: Although DEB theory makes the assumption that food is assimilated at a constant efficiency, there is some evidence that daphnids strive to increase assimilation efficiency at low food concentrations [11], probably by increasing gut residence times. In accordance with Rinke and Vijverberg [12], we assume the assimilation efficiency to be dependent on external food concentration: The scaled functional response then becomes a function of the summed realised assimilation per day and the maximum daily assimilation rate: Reserve dynamic. The scaled amount of reserve U E in an individual is determined by the difference in scaled assimilation f L 2 and the mobilisation flux Sc, with the onset of feeding activity after birth ( : Where under growth conditions ( , with ), the mobilisation flux S c is: Whereby the scaled reserve density, e, is given by: At low environmental food density, the reserve drops below a non-growth boundary ( ).
Thereafter, we calculate S c under the assumption that daphnids do not shrink during starvation and that the reserve is used to pay somatic maintenance costs only: Growth. Growth is considered to be the net result of mobilised reserve that is allocated to soma κ (included in the compound parameter with ), minus the costs for somatic maintenance, thus, the change in structural length is given by: If food is scarce and the amount of reserve does not allow the animal to grow ( ), then .
The animal dry weight includes structure and the mass of reserve and is calculated as a function of the cubed structural length: Maturation and Reproduction. A fraction of the mobilised reserves (1-κ) is allocated to maturation or reproduction. The change in scaled maturity is given by: Once puberty is reached ( ), the maturity level is fixed and mobilised reserves (1-κ) are allocated to the reproduction buffer: At certain time steps, the reproduction buffer is converted into a number of embryos R with a fixed reproduction efficiency k r . The ratio of the energy that can efficiently be used for producing embryos and the cost of a single embryo determines the clutch size: As only whole embryos can be produced, the clutch size is rounded to the lower integer and the remaining energy is kept in the reproduction buffer for the next reproductive event. The DEB theory assumes that initial embryonic reserves are such that the embryo will hatch at the same reserve density as the mother experienced at embryo production. One way to determine the initial reserve would be to test different embryonic reserve levels and to simulate embryo development until the respective maternal reserve density is observed. As this method is computationally expensive, we use an approximation and model embryonic growth ( ) and maturation ( ) by assuming a fixed scaled reserve density, e (set to the maternal e at embryo formation). When embryos surpass scaled maturity at birth ( ), they are released by the mother and a new clutch is formed. At birth, the embryonic reserve level is re-calculated by . Consequently, the reserve level does not change during embryonic development and is only used to determine the clutch size. More effective procedures for the calculation of U E (0) are given by Kooijman [13].
Survival. Two causes of death are considered in the model: ageing and starvation. The basic idea behind the effect of ageing on survival is that damage-inducing compounds are accumulated at a rate proportional to the rate of energy mobilisation and each of the damage-inducing compounds is copied also at a rate proportional to energy mobilization [4]. The process of damage production can be summarised as ageing acceleration q: The resulting hazard rate due to ageing is: The above formulation for ageing assumes that daphnids live longer at low food levels compared to higher ones. However, at some point the food level does not allow the full coverage of maintenance costs and starvation occurs. We assume that at low reserve density, additional damage is caused by some type of physiological disturbance. Modified from Jager et al. [14], this scaled damage (D*) is assumed to accumulate proportional to one minus the scaled reserve density, and is repaired in proportion to the actual damage level with a fixed rate: To link scaled damage to the survival probability, we use the concept of individual tolerance, a special case of the general unified threshold model of survival [14]. In individual tolerance models, the threshold for survival follows a frequency distribution within a population and death is instantaneous for an organism when damage exceeds the individual survival threshold. In accordance with Nyman et al. [15], the threshold is drawn from a log-logistic cumulative distribution function: For a discussion of the starvation model see [16].
Survival probability at time t is calculated based on the individual survival threshold for starvation and the hazard rate for ageing at time t: A Daphnia individual is considered as being dead when the survival probability is below a random number for survival that is drawn from a uniform distribution between 0 and 1 and assigned to the individual at birth.

Implementation
The model is implemented in Delphi Xe2 (Embarcadero Technologies San Francisco, USA, 2011) and is based on discretised forms of the differential equation submodels as described above.

Source code
In the following section the source code of the model, implemented in Delphi XE2, is given. The full implementation including an executable file can be obtained from the authors on request. Please also note, that the executable file does not require the installation of any software, and can be used independent of the Delphi programming environment.