• Energy Research
• natural sciences close
• NL close
• Utrecht University close

AbstractIn a physiologically structured population model (PSPM) individuals are characterised by continuous variables, like age and size, collectively called their i-state. The world in which these individuals live is characterised by another set of variables, collectively called the environmental condition. The model consists of submodels for (i) the dynamics of the i-state, e.g. growth and maturation, (ii) survival, (iii) reproduction, with the relevant rates described as a function of (i-state, environmental condition), (iv) functions of (i-state, environmental condition), like biomass or feeding rate, that integrated over the i-state distribution together produce the output of the population model. When the environmental condition is treated as a given function of time (input), the population model becomes linear in the state. Density dependence and interaction with other populations is captured by feedback via a shared environment, i.e., by letting the environmental condition be influenced by the populations’ outputs. This yields a systematic methodology for formulating community models by coupling nonlinear input–output relations defined by state-linear population models. For some combinations of submodels an (infinite dimensional) PSPM can without loss of relevant information be replaced by a finite dimensional ODE. We then call the model ODE-reducible. The present paper provides (a) a test for checking whether a PSPM is ODE reducible, and (b) a catalogue of all possible ODE-reducible models given certain restrictions, to wit: (i) the i-state dynamics is deterministic, (ii) the i-state space is one-dimensional, (iii) the birth rate can be written as a finite sum of environment-dependent distributions over the birth states weighted by environment independent ‘population outputs’. So under these restrictions our conditions for ODE-reducibility are not only sufficient but in fact necessary. Restriction (iii) has the desirable effect that it guarantees that the population trajectories are after a while fully determined by the solution of the ODE so that the latter gives a complete picture of the dynamics of the population and not just of its outputs.

AbstractIn a physiologically structured population model (PSPM) individuals are characterised by continuous variables, like age and size, collectively called their i-state. The world in which these individuals live is characterised by another set of variables, collectively called the environmental condition. The model consists of submodels for (i) the dynamics of the i-state, e.g. growth and maturation, (ii) survival, (iii) reproduction, with the relevant rates described as a function of (i-state, environmental condition), (iv) functions of (i-state, environmental condition), like biomass or feeding rate, that integrated over the i-state distribution together produce the output of the population model. When the environmental condition is treated as a given function of time (input), the population model becomes linear in the state. Density dependence and interaction with other populations is captured by feedback via a shared environment, i.e., by letting the environmental condition be influenced by the populations’ outputs. This yields a systematic methodology for formulating community models by coupling nonlinear input–output relations defined by state-linear population models. For some combinations of submodels an (infinite dimensional) PSPM can without loss of relevant information be replaced by a finite dimensional ODE. We then call the model ODE-reducible. The present paper provides (a) a test for checking whether a PSPM is ODE reducible, and (b) a catalogue of all possible ODE-reducible models given certain restrictions, to wit: (i) the i-state dynamics is deterministic, (ii) the i-state space is one-dimensional, (iii) the birth rate can be written as a finite sum of environment-dependent distributions over the birth states weighted by environment independent ‘population outputs’. So under these restrictions our conditions for ODE-reducibility are not only sufficient but in fact necessary. Restriction (iii) has the desirable effect that it guarantees that the population trajectories are after a while fully determined by the solution of the ODE so that the latter gives a complete picture of the dynamics of the population and not just of its outputs.

Harmful algal blooms threaten the water quality of many eutrophic and hypertrophic lakes and cause severe ecological and economic damage worldwide. Dense blooms often deplete the dissolved CO2 concentration and raise pH. Yet, quantitative prediction of the feedbacks between phytoplankton growth, CO2 drawdown and the inorganic carbon chemistry of aquatic ecosystems has received surprisingly little attention. Here, we develop a mathematical model to predict dynamic changes in dissolved inorganic carbon (DIC), pH and alkalinity during phytoplankton bloom development. We tested the model in chemostat experiments with the freshwater cyanobacterium Microcystis aeruginosa at different CO2 levels. The experiments showed that dense blooms sequestered large amounts of atmospheric CO2, not only by their own biomass production but also by inducing a high pH and alkalinity that enhanced the capacity for DIC storage in the system. We used the model to explore how phytoplankton blooms of eutrophic waters will respond to rising CO2 levels. The model predicts that (1) dense phytoplankton blooms in low- and moderately alkaline waters can deplete the dissolved CO2 concentration to limiting levels and raise the pH over a relatively wide range of atmospheric CO2 conditions, (2) rising atmospheric CO2 levels will enhance phytoplankton blooms in low- and moderately alkaline waters with high nutrient loads, and (3) above some threshold, rising atmospheric CO2 will alleviate phytoplankton blooms from carbon limitation, resulting in less intense CO2 depletion and a lesser increase in pH. Sensitivity analysis indicated that the model predictions were qualitatively robust. Quantitatively, the predictions were sensitive to variation in lake depth, DIC input and CO2 gas transfer across the air-water interface, but relatively robust to variation in the carbon uptake mechanisms of phytoplankton. In total, these findings warn that rising CO2 levels may result in a marked intensification of phytoplankton blooms in eutrophic and hypertrophic waters.

Harmful algal blooms threaten the water quality of many eutrophic and hypertrophic lakes and cause severe ecological and economic damage worldwide. Dense blooms often deplete the dissolved CO2 concentration and raise pH. Yet, quantitative prediction of the feedbacks between phytoplankton growth, CO2 drawdown and the inorganic carbon chemistry of aquatic ecosystems has received surprisingly little attention. Here, we develop a mathematical model to predict dynamic changes in dissolved inorganic carbon (DIC), pH and alkalinity during phytoplankton bloom development. We tested the model in chemostat experiments with the freshwater cyanobacterium Microcystis aeruginosa at different CO2 levels. The experiments showed that dense blooms sequestered large amounts of atmospheric CO2, not only by their own biomass production but also by inducing a high pH and alkalinity that enhanced the capacity for DIC storage in the system. We used the model to explore how phytoplankton blooms of eutrophic waters will respond to rising CO2 levels. The model predicts that (1) dense phytoplankton blooms in low- and moderately alkaline waters can deplete the dissolved CO2 concentration to limiting levels and raise the pH over a relatively wide range of atmospheric CO2 conditions, (2) rising atmospheric CO2 levels will enhance phytoplankton blooms in low- and moderately alkaline waters with high nutrient loads, and (3) above some threshold, rising atmospheric CO2 will alleviate phytoplankton blooms from carbon limitation, resulting in less intense CO2 depletion and a lesser increase in pH. Sensitivity analysis indicated that the model predictions were qualitatively robust. Quantitatively, the predictions were sensitive to variation in lake depth, DIC input and CO2 gas transfer across the air-water interface, but relatively robust to variation in the carbon uptake mechanisms of phytoplankton. In total, these findings warn that rising CO2 levels may result in a marked intensification of phytoplankton blooms in eutrophic and hypertrophic waters.

AbstractNutrient enrichment can simultaneously increase and destabilise plant biomass production, with co‐limitation by multiple nutrients potentially intensifying these effects. Here, we test how factorial additions of nitrogen (N), phosphorus (P) and potassium with essential nutrients (K+) affect the stability (mean/standard deviation) of aboveground biomass in 34 grasslands over 7 years. Destabilisation with fertilisation was prevalent but was driven by single nutrients, not synergistic nutrient interactions. On average, N‐based treatments increased mean biomass production by 21–51% but increased its standard deviation by 40–68% and so consistently reduced stability. Adding P increased interannual variability and reduced stability without altering mean biomass, while K+ had no general effects. Declines in stability were largest in the most nutrient‐limited grasslands, or where nutrients reduced species richness or intensified species synchrony. We show that nutrients can differentially impact the stability of biomass production, with N and P in particular disproportionately increasing its interannual variability.

AbstractNutrient enrichment can simultaneously increase and destabilise plant biomass production, with co‐limitation by multiple nutrients potentially intensifying these effects. Here, we test how factorial additions of nitrogen (N), phosphorus (P) and potassium with essential nutrients (K+) affect the stability (mean/standard deviation) of aboveground biomass in 34 grasslands over 7 years. Destabilisation with fertilisation was prevalent but was driven by single nutrients, not synergistic nutrient interactions. On average, N‐based treatments increased mean biomass production by 21–51% but increased its standard deviation by 40–68% and so consistently reduced stability. Adding P increased interannual variability and reduced stability without altering mean biomass, while K+ had no general effects. Declines in stability were largest in the most nutrient‐limited grasslands, or where nutrients reduced species richness or intensified species synchrony. We show that nutrients can differentially impact the stability of biomass production, with N and P in particular disproportionately increasing its interannual variability.