Viscoelasticity and other related phenomena are of great importance in the study of mechanical properties of materials, especially biological materials. Certain materials demonstrate some complicated behavior under mechanical tests that cannot be described by a standard linear equation (SLE), mostly due to the shape memory effect during the deformation phase. Recently, researchers have been making use of fractional calculus (FC) in order to probe viscoelasticity of such materials accurately. FC is a powerful tool for modeling complicated phenomena. In this tutorial paper, it is sought to provide clear descriptions of this powerful tool and its techniques and implementation. It is endeavored to keep the details to a minimum while still conveying a good idea of what and how can be done with this powerful tool. The reader will be provided with the basic techniques that are used to solve the fractional equations analytically and/or numerically. More specifically, simulating the shape memory phenomena with this powerful tool will be studied from different perspectives, and some physical interpretations are made in this regard. This paper is also a review of fractional order models of viscoelastic phenomena that are widespread in bioengineering. Thus, in order to show the relationship between fractional models and SLEs, a new fractal system comprising spring and damper elements is considered and the constitutive equation is approximated with a fractional element. Finally, after a brief literature review, two fractional models are utilized to investigate the viscoelasticity of the cell and a comparison is made between the findings and the experimental data from the previous models. Verification results indicate that the fractional model not only matches well with the experimental data but also can be a good substitute for previously used models.

We first present a predator-prey model for two species and then extend the model to three species where the two predator species engage in mutualistic predation. Constant effort harvesting and the impact of by-catch issue are also incorporated. Necessary sufficient conditions for the existence and stability of positive equilibrium points are examined. It is shown that harvesting is sustainable, and the memory concept of the fractional derivative damps out oscillations in the population numbers so that the system as a whole settles on an equilibrium quicker than it would with integer time derivatives. Finally, some possible physical explanations are given for the obtained results. It is shown that the stability requires the memory concept in the model.

Monod’s model describes the growth of microorganisms using a hyperbolic function of extracellular resource concentration. Under fluctuating or limited resource concentrations this model performs poorly against experimental data, motivating the more complex Droop model with a time-varying internal storage pool. We extend the Monod model to incorporate memory of past conditions, adding a single parameter motivated by a fractional calculus analysis. We show how to interpret the memory element in a biological context and describe its connection to a resource storage pool. Under nitrogen starvation at non-equilibrium conditions, we validate the model with simulations and empirical data obtained from lab cultures of diatoms (T. pseudonana and T. weissflogii) and prasinophytes (Micromonas sp. and O. tauri), globally influential phytoplankton taxa. Using statistical analysis, we show that our Monod-memory model estimates the growth rate, cell density and resource concentration as well as the Droop model, while requiring one less state variable. Our simple model may improve descriptions of phytoplankton dynamics in complex earth system models at a lower computational cost than is presently achievable.

Mathematical models of photosynthesis–irradiance relationships in phytoplankton are used to compute integrated water-column photosynthetic rates and predict primary production. Models typically ignore an important phenomenon observed in most experiments: photosynthetic rate remains constant over a range of irradiances before declining due to photoinhibition. Here we develop an approach that captures both photoinhibition and this plateau. We test six new models of photoinhibition and ten more photoinhibition models from the literature against a database of 1808 photosynthesis–irradiance curves exhibiting photoinhibition. The best model improves adjusted R² by 6–37% and reduces RMSE by 15–70% compared to existing models. In the best model, photoinhibition is phenomenologically described by multiplication by a saturating function of the reciprocal of irradiance, simplifies to the widely-used Jassby & Platt photosynthesis–irradiance curve in the absence of photoinhibition, and only requires one new parameter. This photoinhibition parameter identifies the onset of photoinhibition and is the rate of decrease in photosynthetic rate at that irradiance. Simulations show that while parameter values vary up to 40% across existing models, particularly with and without photoinhibition, our representation of photoinhibition does not affect the interpretation or numerical values of parameters compared to the corresponding model without photoinhibition.

Although the macromolecular composition of phytoplankton shapes the nutrition available to marine ecosystems and regulates global biogeochemistry, there are no mechanistic, predictive models for its global distribution. Here, using a cellular allocation model, we simulate phytoplankton allocation to proteins, carbohydrates and lipids in the present day and a warming scenario. Our simulations predict spatial variations consistent with available observations: in nutrient-sufficient, low-light, high-latitude regions, phytoplankton allocate more to nitrogen-rich proteins, while in nutrient-depleted subtropical regions, allocation favours carbohydrates and lipids. Under warming, subtropical phytoplankton increase protein allocation by ~20%, as subsurface populations rich in light-harvesting proteins thrive, whereas high-latitude protein allocation declines by ~15–30% due to warming and relief from light limitation. In situ macromolecular measurements in polar regions show recent trends consistent with our predictions. These results suggest that macromolecular composition responds measurably to changing environmental conditions, reshaping the nutritional landscape at the base of the marine food web.