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But what does a successful invasion mean? Any microorganism that is transported into a new environment or community can be called an invader. We can depict a successful invasion as a four-step process. The first phase is the introduction when alien microbes enter the native community. If an invader can withstand the abiotic pressures of a new environment, we can consider it to be introduced. The acquisition of nutrients in a new environment is not guaranteed yet. Instead, the invaders ability to resist the abiotic conditions of novel environments and then survive the biotic counterpressure of native organisms represents a transition to the second phase of the invasion, which is the establishment. Whatever the habitat in which a microbial invader enters, this phase is where invaders face the strongest biotic resistance from the resident community. If an invader gains access to resources that support its growth, it will pass to the third phase, growing and spreading. Successful invaders may have neutral, beneficial or detrimental impacts in their new habitat. They have the potential to alter the structure of the interactions. For example, if the invader manages to displace a highly interacting microbe that facilitates many interactions, the whole network is prone to collapse. Ecologically speaking, removing a keystone species can result in strong shifts in community structure that severely alter ecosystem functioning. What we want to do is to examine the issue in silico. While there are many and varied experimental researches, purely theoretical ones are much less. In the following, I would like to present what modelling approaches and possibilities exist in a non-exhaustive way. These approaches provide the systematic quantitative characterisation of observed patterns and their underlying mechanisms. A large number of models have been proposed in microbial community ecology. They could be classified as static or dynamic models, depending on if they explicitly include information on how the components of a community interact over time, enabling the study of variations in the properties included in the model. The first classification of dynamic models is based on the level of detail, from the population-level strain composition to detailed models of all metabolic reactions in each cell. A second classification uses the type of algorithm employed to simulate the model. The main types are deterministic models based on differential equations and stochastic models whose behaviour depends on random events. A third criterion serves to classify what is represented in the model. Metabolic models represent the metabolism of the bacteria, often at a molecular level, while in ecological models, strains or species interact, reproduce and change their numbers and behaviours at a cellular level. Models based on Ordinary Differential Equations employ a set of mathematical functions to represent microbial communities at different levels of detail. These deterministic mathematical functions describe the changes in quantity or concentration of different species in time. One of the simplest but most popular microbial community ODE-based models is the generalised Lotka-Volterra equation. Using a small number of parameters such as growth rate and abundances, it evaluates the effect of one microorganism on the growth rate of another. Therefore, it is suitable for assessing pairwise interactions and determining whether they are cooperative or competitive. Together, these interactions encode the community network. Once the parameters have been determined, these can be used to simulate microbial dynamics in silico — for example, to predict community stability and invasibility and to study the role of higher-order interactions within microbial communities. While the gLV model offers a range of applications for studying microbial communities, it also has some limitations. 1) It can only describe pairwise interactions and thus fails to capture modulating effects that a third species may have on an interacting pair. For example, a third species' production of the exchanged metabolite can weaken or strengthen a cross-feeding relationship between two other species. 2) It assumes that populations are homogeneous. 3) Interaction strengths are supposed not to change over time. 4) It does not consider immigration from surrounding communities or environmental effects such as temperature or spatial structure. Another problem is that because of the behaviour of the standard gLV model, we cannot simulate the presence of multistability, thus, when more than one stable state exists in the system. As you can see in this slide, these limitations can be partially overcome by including different extensions to the model. We can simulate higher variability after a perturbation when relaxing the linearity assumption by introducing non-linear growth functions. Extending the gLV model with multiplicative interaction terms, we can exhibit multistability and perturbation-induced switches between alternative stable states. Or, we can take the environmental effects and immigration into account by extending the original model with the correct terms. Returning to the question of invasion, I was curious to see what theoretical results and modelling possibilities exist in this area. There are many exciting and inspiring studies, but due to a lack of time I picked two of them to introduce. The first research I would show is on how cross-feeding within a community affects invasion risk. Incorporating cross-feeding into mathematical models can be computationally challenging since it introduces many additional parameters. We must track concentrations of each metabolite both in the environment and within cells and their exchanges between cells. The author developed a metabolite-explicit model where native microbial taxa interact through both cross-feeding and competition for metabolites. The model progresses through discrete time steps, where the states of the microbes and the external and stored metabolites are updated based on the prior values. Each timestep begins with input metabolites entering the environmental pool when the microbes start to compete for them. Metabolite uptake from the environment is allocated proportionally among taxa following each taxon's demand for the metabolite. Population growth is limited by whatever metabolite is most scarce in the population. The reproducing individuals (those having acquired all necessary metabolites) also excrete one unit of each metabolite in their excretion profile. If these individuals are from taxa participating in cross-feeding, the excreted metabolites are preferentially available to the recipient taxon and directly transferred to the recipient without being available for competitive uptake. Any excreted metabolites that are not part of cross-feeding relationships enter the environmental pools of metabolites. Finally, a certain proportion of the microbes and environmental metabolites are flushed from the system. Well, the invader was introduced after the community of resident taxa equilibrated. After adding the invader to the system, the simulation continued until the community again reached equilibrium. A successful invader changes the abundance of native taxa by introducing additional competition for metabolites. She found that stronger cross-feeding and competition led to much lower invasion risk, as both types of biotic interactions lead to greater metabolite scarcity for the invader. She also evaluated the impact of a successful invader on community composition and structure. The effect of invaders on the native community was most significant at intermediate levels of cross-feeding; at this 'critical' level, successful invaders generally cause decreased diversity and productivity, greater metabolite availability, and reduced quantities of metabolites exchanged among taxa. Secondly, I would like to present another theoretical example where the relationship between the level of auxotrophy and the network's resistance to perturbation is investigated. However, it is not connected with the question of invasion; still, it is an excellent example of how the simple gLV model can be further developed and made suitable to follow the dynamics of metabolites explicitly. They use a population dynamics model to analyse networks of auxotrophic microorganisms that exchange metabolites as extracellular public goods. They describe the dynamics of auxotrophic microbes and the metabolites by this system of ordinary differential equations. This term represents the obligate nature of the interaction between microbes and metabolites. This product ensures that all microbes are going extinct when at least one of the required metabolites is zero. The study aims to identify how the robustness of these networks to ecological disturbance is affected by: 1. The number of metabolic auxotrophies. 2. The topology of the interaction network. 3. The presence of cooperative cross-feeders within the network. By arbitrarily generating bipartite networks and simulating the microbes and metabolites' dynamics until reaching equilibrium, they can get many different topologies and levels of auxotrophy that can affect their robustness against disturbances. They found, for example, that increased degrees of auxotrophy can be detrimental to microbial communities by making them more vulnerable to ecological disturbances. It would be fascinating to do a similar simulation on invasion resistance. For example, what if an invader enters that can provide a resource that has not been present before? Or if it produces nothing but consumes some of the metabolites available? To what extent are communities with different topologies resistant to the emergence of invaders of various degrees of auxotrophy? These and similar questions motivate our current research.