Posters

On the formation of coral reefs

In this study, we present a novel model that successfully replicates the observed formations of coral reefs. Our proposed model is based on partial differential equations and incorporates the wealth of knowledge accumulated over the past decades regarding the physical and ecological interactions occurring at the micro- and mesoscales within these systems. These interactions include clonal growth, facilitation, the uptake of resources by corals, and the supply of these resources by ocean water currents, among others.

Through the mathematical analysis of this model using bifurcations theory, we uncover that the interplay of only a few parameters of the model is sufficient to elucidate the emergence of various reef shapes found in nature. This includes the formation of fringing reefs, closed atolls, and the inner structures within closed atolls. Our findings provide valuable insights into the underlying mechanisms driving the diverse morphologies observed in coral reefs.

Existence, Uniqueness, Boundedness and Long-term Behavior of Solutions to an SIR Model with Intermittent Treatment

We study a dynamic SIR (Susceptible-Infected-Recovered) model with an intermittent treatment term. Results are given for the cases when no components diffuse, only I and R diffuse, and all components diffuse. In each case, we prove global existence and uniform boundedness of solutions, and investigate the long-time behavior.

Evolution of population structured by dietary diversity: cross-diffusion systems

The poster aims to illustrate the analysis of a class of cross-diffusion systems, that is a class of nonlinear parabolic systems with relevant applications in population dynamics. More precisely, I will show the existence and regularity of solutions with an approximation method that uses a system with linear diffusion and additional fast reaction terms. The approximating model describes the evolution of populations, having a different diet. 

We endow the analysis with the study of the linear stability of spatially homogeneous equilibria to prove the occurrence of Turing instability.

Self-organization in phenotypically structured Keller-Segel models 

Linking biodiversity and ecosystem functions in evolved communities on landscapes: A reaction-diffusion approach

Biodiversity loss in recent years has made understanding its impact on the ecosystem vital. Relating biodiversity and ecosystem functions (BEF) would help predict the consequences of species loss and thereby facilitate species conservation efforts. Studies over the years have shown several kinds of BEF relationships, depending on the ecosystem function under consideration: positive, negative, none, unimodal and so on. Focusing on primary production, small-scale experimental studies where biodiversity is artificially maintained have shown a positive relationship with biodiversity. In contrast, large-scale observational studies, which focus on naturally assembled communities, lean towards a unimodal relationship.

Theoretical approaches to understanding the BEF relationships also have been prevalent since the 1970s. Recent advancements in mathematical modelling theory have made it possible to study systems that generate and maintain biodiversity in spatially heterogeneous landscapes. We use a novel eco-evolutionary modelling approach based on reaction-diffusion equations to understand the BEF relationship of evolved resource competitors in spatially structured communities. In these systems, biodiversity arises as a natural response to environmental factors affecting the systems. We hypothesise that these environmental conditions have a role in shaping the BEF relationships.

The autotoxicity effects in oscillatory vegetation patterns

Self-organized vegetation patterns are quite common in many arid and semi-arid areas. A deep study of them may help to predict catastrophic scenarios and identify ecological indicators of climate change, land degradation, and ecosystem resilience [1]. 

Several factors have been thought to be responsible for their emergence, starting from feedback acting in different scales to the interactions between vegetation and the abiotic component of the environments. The main mechanism that has been identified in vegetation patterns is related to the local biomass-water positive feedback at the micro-scale, but it starts to fail in ecosystems in which water availability is not limited [2]. In these cases, another mechanism that may rule the evolution of patterned dynamics is the presence of toxic compounds. In particular, as well-known in agriculture since ancient times, the same plant species cannot grow within the same region after a certain amount of time. This issue is related to the fact that the plant-soil negative feedback leads to an increase in soilborne pathogens, a change in soil microbial communities, and an accumulation of autotoxic compounds [3]. 

In this poster presentation, the main aim is to investigate the role of autotoxicity in vegetation dynamics in sloped semi-arid environments. To this aim, an extension of the 1D Klausmeier model [4, 5] is considered and the occurrence of traveling stripes is studied. Analytical solutions are validated by numerical simulations. 

[1] Rietkerk M. et al.: Evasion of tipping in complex systems through spatial pattern formation. Science 374 (2021). 

[2] M. Rietkerk, J. van de Koppel: Regular pattern formation in real ecosystems. Trends Ecol. Evol. 23, 169–175 (2008). 

[3] Mazzoleni S. et al.: Is plant biodiversity driven by decomposition processes? An emerging new theory on plant diversity. Commun. Ecol. 8, 103–109 (2007). 

[4] Klausmeier C.A.: Regular and irregular patterns in semiarid vegetation. Science 284, 1826–1828 (1999). 

[5] Marasco A. et al.: Vegetation pattern formation due to interactions between water availability and toxicity in plant-soil feedback. Bull. Math. Biol. 76, 2866–2883 (2014).

Hybrid modelling for cancer invasion and metastasis

Cancer cells have the ability to interact with the tumour microenvironment and invade the surrounding tissue by reformulating the extracellular matrix (ECM). The coordinated actions of cancer cells, the ECM, cancer associated fibroblasts (CAFs), and the epithelial to mesenchymal transition (EMT) result to in the invasion of the tissue. In this talk, I will present a multiscale hybrid mathematical model which combines the macroscopic nature of the phenomenon, where solid tumours of epithelial-like cancer cells (ECCs) invade the tissue, as well as the microscopic individual based strategy of mesenchymal-like cancer cells (MCCs). The model consists of partial and stochastic differential equations that describe the evolution of the ECCs and the MCCs while accounting for the transitions between them. Numerical simulations of the proposed model will be presented.

Differential geometric bifurcation problems in pde2path

We present how some differential geometric bifurcation problems can be treated with the Matlab continuation and bifurcation toolbox pde2path. The setup consists of updating a base manifold in each continuation step, and considering the pertinent PDEs for the normal displacement. Examples treated include a Schwarz–P–family and some 4th order biomembrane models. 

Pattern formation in Mechanochemical models

In biology, self-organization and pattern formation happen on many different scales. For example in embryonic development, a homogeneous clump of cells evolves into a complex organism. Alan Turing described this process with a system of reaction-diffusion equations of two chemicals: a short range activator and a long range inhibitor [A. M. Turing, The Chemical Basis of Morphogenesis, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237(641):37-72, 1952]. He found that spatially inhomogeneous perturbations result in a periodic pattern close to the homogeneous equilibrium.

In our research, we include mechanical cues as well as chemical cues, since they also play an important role in tissue deformation [C.M. Nelson et al., Emergent patterns of growth controlled by multicellular form and mechanics. PNAS, 102(33):11594-9, 2005.]. The so-called mechanochemical model describes the deformation of a surface depending on a diffusing morphogen [M. Mercker et al., Modeling and computing of deformation dynamics of inhomogeneous biological surfaces, SIAM Journal of Applied Mathematics, 73(5):1768–1792, 2015]. Here, the curvature of the membrane takes on the role of the inhibitor. We study the possible patterns in this model by inspecting the steady state phase space using tools like numerical simulation and Geometric Singular Perturbation Theory. With this approach, we get a better analytical understanding of a wide range of possible patterns.

Spatial heterogeneity in reaction-diffusion equations

Real-world phenomena are oftentimes influenced by mechanisms that act locally but change the solution behavior globally. I will present three reaction-diffusion models with this system behavior. The spatially heterogeneous mechanisms in a reaction-diffusion model for liver inflammation change the system’s behavior from decaying to the trivial solution towards a stationary heterogeneous steady state. The change can be interpreted as the change between healing and chronic infection courses. A model for invasive mosquitoes has traveling wave solutions that may stop propagating due to local interventions. Third, firebreaks stop wildfires, and the model includes a heterogeneous bulk density as a crucial component. I present analytical results and open questions for predicting the changing behavior.

On the development and application of a general model identification framework to biological systems

With a growing world population and the limited resources available on the planet, the food production is a key issue to address now and in the next decades. In doing this, the development of more effective and safer pesticides will play a key role to maximise the crops yield and meet the food demand, and mathematical models are a tool that can help in the formulation of new products by providing a deeper insight on the phenomena involved in the biological system.

When dealing with biological systems there are several challenges to face, such as i) observability and availability of data [1], ii) the related model identifiability [2], iii) poor controllability and iv) large variability on measured outputs, to name a few. In this work it is proposed to use a general approach to biological systems modelling to tackle these challenges, considering as a case study the foliar uptake of the pesticides [3].

It is presented the approach proposed to model the system, starting from the formulation of candidate compartmental models. The first analysis to be conducted on the models are structural identifiability tests, which can be performed by differential algebra techniques [4]. Along with the structural identifiability, also the practical identifiability of the parameters must be tested by looking at local sensitivity analysis. If multiple models are left after the preliminary analyses, it is proposed to use an artificial neural network-based approach to conduct a screening of the models [5]. Statistical tests on the goodness of fit will also be conducted to discriminate among the models. New experiments can be carefully designed via model-based design of experiments techniques, then performed and the procedure iterated until a single model emerges as the best. The following steps will tackle the parameter estimation and validation of the model.

Results about the structural and practical identifiability tests conducted on compartmental models for the foliar uptake are presented.

References:

[1] Villaverde, A.F., 2018. Observability and Structural Identifiability of Nonlinear Biological Systems. http://arxiv.org/abs/1812.04525

[2] Miao, H., Xia, X., Perelson, A.S., Wu, H., On Identifiability of Nonlinear ODE Models and Applications in Viral Dynamics. SIAM review. Society for Industrial and Applied Mathematics, vol 53, pp. 3-39, 2011. DOI: 10.1137/090757009

[3] Wang, C.J., Liu, Z.Q., Foliar uptake of pesticides – Present status and future challenge. Pesticide Biochemistry and Physiology, vol 87, pp 1-8, 2007. DOI: 10.1016/j.pestbp.2006.04.004

[4] Chis, O.T., Banga, J.R., Balsa-Canto, E., Structural identifiability of systems biology models: A critical comparison of methods, PLoS ONE, vol 6, 2011. DOI: 10.1371/journal.pone.0027755

[5] Quaglio, M., Roberts, L., Bin Jaapar, M.S., Fraga, E., Dua, V., Galvanin, F., An artificial neural network approach to recognise kinetic models from experimental data, Computers and Chemical Engineering, vol 135, 2020. DOI: 10.1016/j.compchemeng.2020.106759

Acknowledgements:

Project conducted in collaboration with Federica Cattani (Syngenta) and Dr. Federico Galvanin.

I want to thank UCL Chemical Engineering Department and Syngenta for the financial support.

The reconstruction of parameters in diffusive processes as a model of measurement

The reconstruction of initial values and material parameters in a diffusive partial differential equation from data for propagated time instants has a strong connection to quantities in a measurement process. For example, reconstructing a temperature distribution based on time-delayed discrete data is a measurement problem with loss of information or rather a loss of sensitivity of the information. We present a model of a general measurement procedure, which allows us to investigate the relation between the concept to be measured and the measurement result and to quantify measurement uncertainties. 

Existence and properties of travelling fronts in a nonautonomous 2 component singularly PDE

The starting point is a well understood model given by a  reaction diffusion equation with coefficients U and V, where U satisfies an Allen Cahn equation and V a linear diffusion equation, the coupling is linear in both ways (effect of V on U and vice versa). The effect of V on U is assumed to be small which makes the RDE singularly perturbed. Introducing a comoving frame one can construct travelling front solutions by using geometric singular perturbation theory.

The main novelty of my research is to introduce a space dependent coefficient in the equation for V. As a result the PDE becomes nonautonomous and the comoving frame approach does no longer work. Indeed due to loss of spacial symmetry we see (from simulations) that the front no longer travels at a constant speed, the shape of the front also does not stay exactly the same, and moreover, the equilibrium states to which the front is asymptotic are no longer constant but functions of space. Hence in comoving coordinates these background states become travelling which is the main problem why this method does not work. 

Our main results are developing of new techniques to prove the existence of fronts in such a setup and to study how the speed of the front behaves in time (which as mentioned is no longer constant) by means of a delay differential equation.