Abstract. This paper is continuation of previous work by the present author, where explicit formulas for the eigenvalues associated with several tridiagonal matrices were given. In this paper the associated eigenvectors are calculated explicitly. As a consequence, a result obtained by WenChyuan Yueh and independently by S. Kouachi, concerning the eigenvalues and in particular the corresponding eigenvectors of tridiagonal matrices, is generalized. Expressions for the eigenvectors are obtained that differ completely from those obtained by Yueh. The techniques used herein are based on theory of recurrent sequences. The entries situated on each of the secondary diagonals are not necessary equal as was the case considered by Yueh. Key words. Eigenvectors, Tridiagonal matrices.AMS subject classifications. 15A18. Introduction.The subject of this paper is diagonalization of tridiagonal matrices. We generalize a result obtained in [5] concerning the eigenvalues and the corresponding eigenvectors of several tridiagonal matrices. We consider tridiagonal matrices of the formwhere {a j } n−1 j=1 and {c j } n−1 j=1 are two finite subsequences of the sequences {a j } ∞ j=1 and {c j } ∞ j=1 of the field of complex numbers C, respectively, and α, β and b are complex numbers. We suppose thatwhere d 1 and d 2 are complex numbers. We mention that matrices of the form (1) are of circulant type in the special case when α = β = a 1 = a 2 = ... = 0 and all the entries on the subdiagonal are equal. They are of Toeplitz type in the special case when α = β = 0 and all the entries on the subdiagonal are equal and those on the superdiagonal are also equal (see U. Grenander and G. Szego He has calculated, in this case, the eigenvalues and their corresponding eigenvectorswhere θ k = where d is a complex number. We have proved that the eigenvalues remain the same as in the case when the a i 's and the c i 's are equal but the components of the eigenvector u (k) (σ) associated to the eigenvalue λ k , which we denote by uwhere θ k is given by formula d 2 sin (n + 1) θ k − d (α + β) sin nθ k + αβ sin (n − 1) θ k = 0, k = 1, ..., n.Recently in S. Kouachi [6], we generalized the above results concerning the eigenvalues of tridiagonal matrices (1) satisfying condition (2), but we were unable to calculate the corresponding eigenvectors, in view of the complexity of their expressions. The

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In the current manuscript we comment on (Misra and Babu, Model Earth Syst Environ 2(1):1-11, 2016), where two novel five-species ODE models are proposed and analyzed, in order to investigate the population dynamics of a three-species food chain, in a polluted environment. It is shown in Misra and Babu (Model Earth Syst Environ 2(1):1-11, 2016) that under certain restrictions on the parameters, the models have bounded solutions for all positive initial conditions. Furthermore, a globally attracting set is explicitly constructed for initial conditions in R 5 þ. We prove these results are not true. To the contrary, solutions to these models can blow-up in finite time, even under the parametric restrictions derived in Misra and Babu (Model Earth Syst Environ 2(1):1-11, 2016), for sufficiently large initial conditions. We provide both analytical proofs and numerics to confirm our results.

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As we all know, the use of heroin and other drugs in Europe and more specifically in Ireland and the resulting prevalence are well documented. A huge population is still dying using heroin every day. This may happen due to, several reasons like, excessive use of painkiller, lack of awareness etc. It has also inspired mathematical modelers to develop dynamical systems predicting the use of heroin in long run. In this work, the effect of heroin in Europe has been discussed by constructing a suitable mathematical model. Our model describes the process of treatment for heroin users by consolidating a sensible utilitarian structure that speaking to the restricted accessibility of treatment. In the treatment time frame, because of the discretion of the medication clients, some kind of time delay called immunity delay might be found. The effect of immunity delay on the system’s stability has been examined. The existence of positive solution and its boundedness has been established. Also, the local stability of the interior equilibrium point has been studied. Taking the immunity delay as the key parameter, the condition for Hopf-bifurcation has been studied. Using normal form theory and center manifold theorem, we have likewise talked about the direction and stability of delay induced Hopf-bifurcation. The corresponding reaction diffusion system with Dirichlet boundary condition has been considered and the Turing instability has been studied. Obtained solutions have also been plotted by choosing a suitable value of the parameters as the support of our obtained analytical results.

In Debnath et al. (2019), a tritrophic food chain model subject to a Allee effect on the prey growth and Crowley-Martin senses functional response between intermediate predator and top predator, with a top predator of sexually reproductive type is considered. It is claimed that under certain restrictions on the parameter space, the model has bounded solutions for all positive initial conditions, and is dissipative. We show that this is not true. In particular, solutions to the model can blow-up in finite time, even under the restrictions derived in Debnath et al.(2019), for sufficiently chosen initial data. We derive a new extinction boundary for the system. We also conjecture on the effect of the Allee threshold on the blow-up dynamics in the model. All of our results are validated via numerical simulations. K E Y W O R D SAllee effect, finite time blow up, three species food chain

In order to study the asymptotic behavior, several authors claimed global existence in time of solutions to a tritrophic food chain models following a modied Leslie-Gower formulation considering the interactions between three species: a generalist top predator depredating on a middle predator, that in turn is depreds a prey. To the contrary it is shown nite time blow-up in such models can occur. We show in this work that blow up in nite time persists even when the intermediate (middle) predator is abscent to the contrary to what it is claimed by Kundu and Patra (2022, [13]). It is shown under some restrictions on the parameters, the model has bounded solutions for all positive initial conditions. We show that this is not true. Solutions to the model can blow up in nite time, for initial data suciently large, even under the restrictions derived by the authors. We can show same results even for small initial data but we concentrate our proofs for the rst case. We also show similar results for the spatially extended system. We illustrate all our results through numerical simulations.