In the second part, we explore existence of positive solutions, persistence, and boundedness of solutions for the Nicholson blowflies model with delayed mortality term $-\delta N(h(t))$. Two global stability tests for the positive equilibrium are obtained based on a linearized global stability method, reducing stability of a non-linear model to a specially constructed linear equation. The first test extends the absolute stability result to the case of delayed mortality, and the second one is delay-dependent.

For the model with a simple microorganism, a refinement of the previous analysis provides the uniform persistence, as well as the global attractivity of a positive periodic solution.

These results generalize and enhance recent achievements in the literature, see [1,2].

[1] T. Faria, Periodic solutions for a delayed competitive chemostat model with periodic nutrient input and rate, Nonlinearity (2025) (to appear).

[2] T. Faria, J.G. Mesquita, A competitive chemostat model with time-dependent delays (submitted).

We derive the immune-inactivated reproduction number $\mathcal{R}_0$ and the immune-activated reproduction number $\mathcal{R}_1$ to identify conditions for the persistence of the virus within the host.

When \(\mathcal{R}_0 < 1\), the virus-free steady state \(E_0\) is globally asymptotically stable, indicating that the virus is eventually cleared.

When \(\left(1 - \frac{1}{\mathcal{R}_0}\right)\frac{s \omega_2}{d_3} < 1 < \mathcal{R}_0 \), the CTL-inactivated infection steady state \(E_1^*\) is locally stable. Under these circumstances, we establish criteria for the global stability of \(E_1^*\), which implies the virus and infected cells persist, resulting in a chronic infection without activating a CTL immune response.

When \(\mathcal{R}_1 > 1\), the CTL-activated infection steady state \(E_2^*\) becomes locally stable, and condition for the global stability of \(E_2^*\) is provided. In this case, the infection remains chronic, sustained by an active CTL immune response.

In this talk, we will examine the impact and interplay of harvesting deduction and the choice of diffusion strategy on competition outcomes for two-resource sharing species by analyzing the principal eigenvalues of the system. Previous studies on the impact of harvesting required it to be spatially identical to the growth rate. Here, we consider a more general form of harvesting and we will develop a spatial arrangement of harvesting, dependent on diffusion strategy, that can push two populations to coexist. Further, we will also examine the impact of perturbations to harvesting policies and diffusion strategies, and show that very small perturbations do not affect the populations' ability to coexist. Finally, we will consider a situation with two invading species, where the invader chooses a diffusion strategy that mimics the spatial distribution of the resident species. We will show that when the invader has a higher carrying capacity, it is guaranteed to have a successful invasion. However, numerical simulations show that invasion may be successful even without an advantage in carrying capacity.

In recent decades, this perspective has been useful in control of PDEs, applying results on stabilization of delay systems. Furthermore, the time delay inherent in the transmission of electromagnetic and gravitational waves allows for the asymptotic study of such systems as neutral equations.

In addition to the theoretical results, we numerically explore the viral dynamics beyond equilibrium stability. Our simulations reveal that delays in CTL recruitment can induce transitions between stable equilibria and sustained oscillatory behavior in the viral load, and vice versa. Furthermore, we compare the relative contributions of virus-to-cell and cell-to-cell infection modes to the overall infection level and identify key parameters that influence this balance.

Finally, we demonstrate how Filippov control methods can be employed in antiretroviral therapy to achieve desirable treatment outcomes.