[@ShZh0 Proposition 3.1 (1)].

\(2) follows from the arguments in [@ShZh0 Proposition 3.1 (2)].

\(3) We consider the case $i=1$. Other cases ca be proved similarly.

Note that $u_1(t, x; s, u_0, m_1^2)$ is a supersolution of with $m_1$ being replaced by $m_1^1$. Then by (1), $$u_1(t, \cdot; s, u_0, m_1^1)\leq  u_1(t, \cdot; s, u_0, m_1^2)  \text{ for all } t>s.$$

For simplicity in notation, put $\Phi_i(T; m_i)=\Phi_i(T, 0; m_i) (i=1, 2, 3)$, and let $r(\Phi_i(T;m_i))$ be the spectral radius of $\Phi_i(T, 0; m_i)$.

Basic properties of principal spectrum points/principal eigenvalues
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Our objective in this subsection is to study some basic properties of the principal spectrum point of the regular eigenvalue problem .

Recall that in , $\mathcal (\mathcal L_i+\lambda_i m_i)u=-\p_t u-B_i u+K_iu+\lambda_i m_i u$, and we may use $\mu_i^n(\mathcal L_i+\lambda_i m_i)$ or $\mu_i^n(\lambda_i)$ to denote the principal spectrum point of , if no confusion occurs. And when $\lambda_i=0$, we use $\mu_i^n(0)$ to denote the principal $\mathcal L_i$.

\[spectrum-sol-operator\] For $1\leq i\leq 3$, $\mu_i^n (\mathcal L_i+\lambda_i m_i)=\frac{\ln r(\Phi_i(T; m_i))}{T}$.

It follows from [@RaSh Proposition 3.10].

\[spectrum-limit\] Let $1\le i\le 3$. Given any $u_0\in X_i^+\setminus\{0\}$, $$\mu_i^n(\mathcal{L}_i+\lambda_i m_i)=\lim_{t\to\infty}\frac{\ln\|\Phi_i(t,0;m_i)u_0\|}{t}.$$

It follows from the arguments in [@HuShVi Proposition 2.5 and Theorem 3.2].

\[spectrum-autonomous\] 