*2.2. Case of Parameter γ 'Middle' Values*

We restrict ourselves to considering the boundary value problem (11), (12) in the case when the parameters *h* and *T* coincide, namely *h* = *T*, which is interesting for mathematical ecology problems. Then, the characteristic Equation (22) splits into two:


Moreover, each root of the last equation is repeated infinitely many times. Let the inequality *rT* < *π*/2 hold. Thus, the size of each isolated population does not oscillate in the positive equilibrium state neighborhood. We assume that the problem of the stationery stability in (11), (12) has a critical case: the relations

$$
\sigma\_0 (1 + \gamma\_0) T\_0 = \frac{\pi}{2} \tag{38}
$$

hold for some *r* = *r*0, *γ* = *γ*<sup>0</sup> and *T* = *T*0. From here, the linear boundary value problem (21) has infinitely many periodical solutions *uk*(*t*, *x*) = exp(*ikx* + *iω*0*t*), *k* = 1, 2, ..., *ω*<sup>0</sup> = *π*(2*T*0)−<sup>1</sup> = *r*0(1 + *γ*0). This implies that under the condition *M*(*ξ*(*x*)) = 0, the function *u*0(*t*, *x*) = *ξ*(*x*) exp(*iω*0*t*) is also a solution of (21). We introduce a small parameter *μ* : 0 < *μ* 1. Let

$$r = r\_0 + \mu r, \quad \gamma = \gamma\_0 + \mu \gamma, \quad T = T\_0 + \mu T\_1$$

in (11), (12). We seek solutions to this boundary value problem in the form of formal asymptotic series

$$u(t, \mathbf{x}, \boldsymbol{\mu}) = \mu^{1/2} \left( \tilde{\boldsymbol{\xi}}(\mathbf{r}, \mathbf{x}) \exp\left(i\omega\_0 t\right) + \overline{c\boldsymbol{\mathcal{E}}} \right) + \mu u\_2(t, \mathbf{r}, \mathbf{x}) + \mu^{3/2} u\_2(t, \mathbf{r}, \mathbf{x}) + \dots,\tag{39}$$

where *τ* = *μt*, *ξ*(*τ*, *x*) is the unknown amplitude, and the functions *uj*(*t*, *τ*, *x*) are 2*π*/*ω*0 periodic with respect to *t* and 2*π*-periodic with respect to *x*. The key condition is that the function *ξ*(*τ*, *x*) has a zero mean with respect to the spatial variable:

$$M(\mathfrak{f}(\mathfrak{r}, \mathfrak{s})) = 0. \tag{40}$$

We insert (39) into (11). First, we perform standard actions to obtain the equation for *u*2:

$$\begin{split} \frac{\partial \mu\_2}{\partial t} &= -r\_0 \mu\_2 (t - T, \mathbf{x}) + r\_0 \gamma\_0 \left[ M \{ \mu\_2 (t - T, \mathbf{s}) \} - \mu\_2 (t - T, \mathbf{x}) \right] - \\ &\quad r\_0 (1 + \delta \gamma\_0) \xi^2 (\mathbf{r}, \mathbf{x}) \exp(-i\omega\_0 T\_0 + 2i\omega\_0 t) + \overline{c\mathbf{c}}. \end{split} \tag{41}$$

We seek the solution of (41) in the form

$$
\mu\_2(t,\tau,\mathbf{x}) = \mu\_{20}(t,\tau)\exp(2i\omega\_0 t) + \overline{\varepsilon}\overline{\varepsilon} + \mu\_{21}(t,\tau,\mathbf{x})\exp(2i\omega\_0 t) + \overline{\varepsilon}\overline{\varepsilon}
$$

and *M*(*u*21(*t*, *τ*,*s*)) = 0. Then,

$$\begin{array}{rcl} \mu\_{20}(t,\tau) &=& \mathbb{C}\_{1}M(\tilde{\xi}^{2}(\tau,s)),\\ \mu\_{21}(t,\tau,\mathbf{x}) &=& \mathbb{C}\_{2}\{\tilde{\xi}^{2}(\tau,\mathbf{x}) - M(\tilde{\xi}^{2}(\tau,s))\},\\ \mathbb{C}\_{1} &=& -\left(2i\omega\_{0} + r\_{0}\exp\left(-i\omega\_{0}T\_{0}\right)\right)^{-1}r\_{0}(1+\delta\gamma\_{0})\exp\left(-i\omega\_{0}T\_{0}\right),\\ \mathbb{C}\_{2} &=& -\left(2i\omega\_{0} + r\_{0}(1+\gamma\_{0})\exp\left(-i\omega\_{0}T\_{0}\right)\right)^{-1}r\_{0}(1+\delta\gamma\_{0})\exp\left(-i\omega\_{0}T\_{0}\right). \end{array}$$

At the next step, we collect the coefficients at *μ*3/2 in the formal identity and obtain the equation for *u*<sup>3</sup> in the form

$$\frac{\partial \mu\_3}{\partial t} = -r\_0 \mu\_3(t - T, \mathbf{x}) + r\_0 \gamma\_0 \left[ M \left( \mu\_3(t - T, s) \right) - \mu\_3(t - T, \mathbf{x}) \right] + \dotsb$$

$$D\_1(\tau, \mathbf{x}) \exp(i\omega\_0 t) + \overline{c\overline{c}} + D\_3(\tau, \mathbf{x}) \exp(3i\omega\_0 t) + \overline{c\overline{c}}.$$

The *D*3(*τ*, *x*)-independent in the indicated class of functions solvability condition of this equation is the validity of the equality

$$D\_1(\mathfrak{r}, \mathfrak{x}) - M(D\_1(\mathfrak{r}, \mathfrak{x})) = 0.$$

We take into account the explicit form of *D*1(*τ*, *x*) and obtain from this equality the boundary value problem as the QNF to find *ξ*(*τ*, *x*):

$$\frac{\partial \tilde{\xi}}{\partial \pi} = b\_1 \tilde{\xi} + \beta\_1 \tilde{\xi} |\tilde{\xi}^2| + \beta\_2 \tilde{\xi} M(\tilde{\xi}^2),\tag{42}$$

$$\xi(\tau, \mathbf{x} + 2\pi) \equiv \xi(\tau, \mathbf{x}), \quad M(\xi(\tau, \mathbf{s})) = 0. \tag{43}$$

The formulas

$$\begin{array}{rcl} b\_{1} &=& \left(1 + \frac{\pi^{2}}{4}\right)^{-1} \Big[ \left(\frac{\pi}{2} + i\right) (r\_{1}(1+\gamma\_{0}) + \gamma\_{1}r\_{0}) + r\_{0}^{2}(1+\gamma\_{0}^{2})T\_{1} \left(1 - i\frac{\pi}{2}\right) \Big] / \\\\ \beta\_{1} &=& -r(1+\delta\gamma\_{0}) \Big(\exp(-2i\omega\_{0}T\_{0}) + \exp(i\omega\_{0}T\_{0})\Big) \mathbb{C}\_{2} \left(1 + i\frac{\pi}{2}\right)^{-1} \\\\ \beta\_{2} &=& -r\Big[ (1+\delta\gamma\_{0}) \Big(\exp(-2i\omega\_{0}T\_{0}) + \exp(i\omega\_{0}T\_{0})\Big) \Big(\mathbb{C}\_{1} - \mathbb{C}\_{2}\big) + \\\\ & & \delta\gamma\_{0} \exp(-2i\omega\_{0}T\_{0}) \mathbb{C}\_{1} \Big] \Big(1 + i\frac{\pi}{2}\Big)^{-1} \end{array}$$

hold for the coefficients in (42).

Let us sum things up.

**Theorem 2.** *Let condition* (38) *be satisfied, and the boundary value problem* (42)*,* (43) *has the bounded solution ξ*(*τ*, *x*) *as τ* → ∞*. Then, the function*

$$
\mu(t, \mathbf{x}, \mu) = \mu^{1/2} \left( \mathfrak{f}(\mathbf{r}, \mathbf{x}) \exp \left( i \omega\_0 t \right) + \overline{c} \overline{c} \right) + \mu \mu\_2(t, \mathbf{r}, \mathbf{x}),
$$

*satisfies the boundary value problem* (11)*,* (12) *up to O*(*μ*3/2)*.*

Periodic with respect to *t* and 2*π*-periodic piecewise continuous with respect to *x* solutions to the boundary value problem (11), (12) could also be determined in explicit form. However, it is not considered in this paper. We only refer to the paper [29]. In a similar situation, families of step-like solutions to the QNF are constructed, its stability is studied, and a comparison with experimental data is performed.
