DSolve did not give solution:
ClearAll[x, y, z, gamma, t, a, b, c, d, e];
eq1 = x'[t] == -a x[t] - b x[t] y[t] - d x[t] z[t];
eq2 = y'[t] == a x[t] - b y[t] + c z[t] - d x[t] y[t];
eq3 = z'[t] == b y[t] - c z[t] - c x[t] z[t];
eq4 = gamma'[t] == d x[t] y[t] + e x[t] z[t];
DSolve[{eq1, eq2, eq3, eq4}, {x[t], y[t], z[t], gamma[t]}, t]
But NDsolve can solve them. You need to supply numerical values for all the parameters and initial conditions also.
ClearAll[x, y, z, gamma, t, a, b, c, d, e];
a = 1; b = 2; c = 3; d = 4; e = 5;
eq1 = x'[t] == -a x[t] - b x[t] y[t] - d x[t] z[t];
eq2 = y'[t] == a x[t] - b y[t] + c z[t] - d x[t] y[t];
eq3 = z'[t] == b y[t] - c z[t] - c x[t] z[t];
eq4 = gamma'[t] == d x[t] y[t] + e x[t] z[t];
NDSolve[{eq1, eq2, eq3, eq4, x[0] == 1, y[0] == 0, z[0] == 0,
gamma[0] == 0}, {x[t], y[t], z[t], gamma[t]}, {t, 0, 10}]
